Question 1

Rationalize and simplify the surd expression: If (5 + 2√3) / (7 + 4√3) = a + b√3, determine the exact values of a and b.

Accepted Answer

Introduction / Context: Converting a fraction with surds into the form a + b√3 requires rationalizing the denominator. By multiplying numerator and denominator by the conjugate of the denominator, we eliminate the surd from the denominator and then collect rational and irrational parts separately.Given Data / Assumptions:Original expression: (5 + 2√3) / (7 + 4√3).Target form: a + b√3 where a and b are rational numbers.Conjugate of (7 + 4√3) is (7 − 4√3).Concept / Approach: Multiply numerator and denominator by the conjugate (7 − 4√3). Use (p + q√3)(p − q√3) = p^2 − 3q^2 to remove the surd from the denominator. Then expand the numerator and separate terms with and without √3 to identify a and b.Step-by-Step Solution:Compute denominator: (7 + 4√3)(7 − 4√3) = 49 − 3·(4^2) = 49 − 48 = 1.Compute numerator: (5 + 2√3)(7 − 4√3) = 35 − 20√3 + 14√3 − 8·3.Simplify numerator: 35 − 6√3 − 24 = 11 − 6√3.Thus, (5 + 2√3)/(7 + 4√3) = 11 − 6√3 ⇒ a = 11, b = −6.Verification / Alternative check: Because the rationalized denominator equals 1, the product in the numerator is already the final simplified value. Quick numerical check: evaluate both sides approximately to confirm consistency.Why Other Options Are Wrong: The signs of a and b are determined by the cross terms (−20√3 + 14√3 = −6√3). Any option with b positive or wrong a-value ignores this cancellation.Common Pitfalls: Forgetting that (√3)^2 = 3 when expanding, or using the wrong conjugate sign leading to an incorrect denominator.Final Answer: a = 11, b = -6

Question 2

Evaluate the power tower with surds: Compute the exact value of [(√2)^(√2)]^(√2). Choose the best classification of the result.

Accepted Answer

Introduction / Context: Exponent laws remain valid for positive real bases and real exponents: (a^b)^c = a^(b*c). This problem applies that idea to a surd base √2 and two surd exponents √2. The goal is to simplify the nested exponents and then classify the final value in the standard number hierarchy.Given Data / Assumptions:Expression: [(√2)^(√2)]^(√2).Base a = √2 > 0, so exponent laws for real exponents apply.All operations are real-valued.Concept / Approach: Use the law (a^b)^c = a^(b*c). Multiplying the two √2 exponents gives √2 * √2 = 2. Thus the entire tower collapses to (√2)^2, which is an exact integer. We then identify its set membership: integer, natural, rational, real.Step-by-Step Solution:Start with E = [(√2)^(√2)]^(√2).Apply exponent law: E = (√2)^(√2 * √2) = (√2)^(2).Evaluate: (√2)^2 = 2.Verification / Alternative check: Compute numerically: √2 ≈ 1.4142. Then (√2)^(√2) ≈ 1.4142^1.4142 ≈ 1.632..., and raising that to √2 returns exactly 2 by the algebraic identity used.Why Other Options Are Wrong: The value 2 is an integer and a natural number (in the usual positive-integer convention). Therefore it is not “non-integer rational” nor “irrational.”Common Pitfalls: Believing that a surd base automatically produces an irrational result after exponentiation. Here, the structure of exponents forces an exact integer after simplification.Final Answer: a natural number

Question 3

Simplify the compound monomial quotient: Simplify (6 a^(-2) b c^(-3) / (4 a b^(-3) c^2)) ÷ (5 a^(-3) b^2 c^(-1) / (3 a b^(-2) c^3)) and express the final answer using positive/negative exponents as needed.

Accepted Answer

Introduction / Context: This is a standard indices exercise involving negative exponents and division of monomials. The safest route is to simplify each fraction separately using index rules, then divide them. Carefully add or subtract exponents when multiplying or dividing like bases, and reduce numerical coefficients.Given Data / Assumptions:First factor: (6 a^(-2) b c^(-3)) / (4 a b^(-3) c^2).Second factor: (5 a^(-3) b^2 c^(-1)) / (3 a b^(-2) c^3).All variables represent nonzero real numbers (so divisions are valid).Concept / Approach: Use three rules repeatedly: a^m / a^n = a^(m−n); (ab)^m = a^m b^m; and handle coefficients separately. Simplify each fraction to a single monomial, then compute the quotient of those monomials by subtracting exponents and dividing coefficients.Step-by-Step Solution:First fraction: coefficients 6/4 = 3/2.a-exponent: (−2) − (1) = −3 ⇒ a^(−3).b-exponent: 1 − (−3) = 4 ⇒ b^4.c-exponent: (−3) − 2 = −5 ⇒ c^(−5).So F1 = (3/2) a^(−3) b^4 c^(−5).Second fraction: coefficients 5/3.a-exponent: (−3) − (1) = −4 ⇒ a^(−4).b-exponent: 2 − (−2) = 4 ⇒ b^4.c-exponent: (−1) − 3 = −4 ⇒ c^(−4).So F2 = (5/3) a^(−4) b^4 c^(−4).Quotient: F1 ÷ F2 = (3/2)/(5/3) * a^(−3 − (−4)) * b^(4−4) * c^(−5 − (−4)).Compute: (3/2)*(3/5) = 9/10; a-exponent = 1; b-exponent = 0; c-exponent = −1.Final: (9/10) a c^(−1) = (9/10) · a / c.Verification / Alternative check: Convert to positive-exponent form: 9a/(10c). Any equivalent representation matches the simplified result.Why Other Options Are Wrong: Options with c, c^2, or c^(−3) come from exponent slips; a^(−1) would require subtracting in reverse.Common Pitfalls: Mixing the direction of subtraction for exponents during division and forgetting to invert the second coefficient when dividing fractions.Final Answer: 9/10 · a c^-1

Question 4

Use exponent rules to condense powers of 3: Evaluate 27^3 × 3^4 ÷ 3^10 and choose the correct value.

Accepted Answer

Introduction / Context: This question tests consolidation of exponents using the same base. The key is to rewrite 27 as a power of 3 and then apply the rules for multiplying and dividing powers with the same base: add exponents when multiplying and subtract when dividing.Given Data / Assumptions:Expression: 27^3 × 3^4 ÷ 3^10.Recall: 27 = 3^3.Exponent rules: a^m * a^n = a^(m+n), and a^m / a^n = a^(m−n).Concept / Approach: Express all terms as powers of 3. Then use exponent laws to combine them into a single power of 3. Finally, evaluate that power or identify the corresponding integer.Step-by-Step Solution:Rewrite 27^3 as (3^3)^3 = 3^(9).Multiply: 3^9 × 3^4 = 3^(9+4) = 3^13.Divide by 3^10: 3^13 ÷ 3^10 = 3^(13−10) = 3^3.Evaluate: 3^3 = 27.Verification / Alternative check: You can also pair 3^4 ÷ 3^10 = 3^(−6) and then multiply 27^3 * 3^(−6) = 3^9 * 3^(−6) = 3^3 = 27.Why Other Options Are Wrong: 9 and 81 are 3^2 and 3^4 respectively—off by exponent arithmetic; 1/27 corresponds to 3^(−3), which would arise from reversing the subtraction; 3 would require exponent 1.Common Pitfalls: Forgetting to convert 27 into 3^3, or subtracting exponents in the wrong order when dividing.Final Answer: 27

Question 5

Combine exponents with a common base: If 10^(2/5) × 10^(8/5) = 10^n, determine the value of n.

Accepted Answer

Introduction / Context: This quick test checks your fluency with adding exponents when multiplying like bases. The base 10 is common to both terms, so the product is another power of 10 with an exponent equal to the sum of the original exponents.Given Data / Assumptions:Bases are the same: 10.Exponents: 2/5 and 8/5.Property: a^m * a^n = a^(m + n) for any real exponents m and n.Concept / Approach: Add the fractional exponents: 2/5 + 8/5 = 10/5. Then simplify the fraction to get an integer exponent. The result is a simple power of 10 that can be named or computed numerically if needed.Step-by-Step Solution:n = (2/5) + (8/5) = 10/5 = 2.Therefore 10^(2/5) × 10^(8/5) = 10^2 = 100.Verification / Alternative check: Evaluate numerically: 10^(0.4) × 10^(1.6) = 10^(2.0) by exponent addition, confirming n = 2.Why Other Options Are Wrong: 3, 4, 6, 10 reflect mis-additions or multiplications of exponents instead of addition.Common Pitfalls: Multiplying exponents (incorrect) instead of adding them when bases are equal.Final Answer: 2

Question 6

Apply the laws of indices to simplify variables and coefficients: Compute 12 x^2 y^4 z^3 ÷ (2 x y^2 × 3 y z^2) and express the answer in simplest monomial form.

Accepted Answer

Introduction / Context: This is a basic practice question on simplifying products and quotients of monomials. The idea is to reduce the numerical coefficient and then subtract exponents of like bases appearing in numerator and denominator.Given Data / Assumptions:Numerator: 12 x^2 y^4 z^3.Denominator: (2 x y^2) × (3 y z^2) = 6 x y^3 z^2.All variables x, y, z are nonzero real numbers.Concept / Approach: Divide coefficients: 12 ÷ 6 = 2. For variables, subtract exponents according to a^m / a^n = a^(m−n). Do this separately for x, y, and z. The result should be a single monomial with positive exponents.Step-by-Step Solution:Coefficient: 12 / 6 = 2.x: x^2 / x^1 = x^(2−1) = x.y: y^4 / y^3 = y^(4−3) = y.z: z^3 / z^2 = z^(3−2) = z.Final simplified form: 2 x y z.Verification / Alternative check: Substitute simple nonzero values (e.g., x = y = z = 1) to see that both sides equal 2, supporting the algebra.Why Other Options Are Wrong: xyz and 1/(xyz) ignore the coefficient change and/or exponent subtraction; “2 xyz” is equivalent in spacing to 2xyz (correct), but other distractors like x^2 y z reflect incorrect exponent arithmetic.Common Pitfalls: Forgetting to combine the two factors in the denominator first and mishandling exponent subtraction order when dividing.Final Answer: 2xyz

Question 7

If $\sqrt{3^n} = 729$, then the value of $n$ is

Accepted Answer

Concept & Formula To solve equations with the unknown variable in the exponent, express both sides of the equation using the exact same base so the exponents can be directly equated. (a^x)^y = a^{xy} Step-by-Step Solution • Given: \sqrt{3^n} = 729 • Calculation: Express the square root as a fractional exponent on the left side: (3^n)^{(1/2)} = 3^{(n/2)} • Express the right side (729) as a power of the base 3: 729 = 9 × 81 = 3^2 × 3^4 = 3^6 • Rewrite the full equation with the matching bases: 3^{(n/2)} = 3^6 • Since the bases are identical (3), their exponents must be equal: (n/2) = 6 • Multiply by 2 to solve for n: n = 12 Exam Strategy & Shortcut If you know your powers of 3, you know 3^6 = 729. Since the left side is a square root, the internal power n must be double the required exponent to survive the root process. Therefore, n = 6 × 2 = 12. Common Pitfall Equating n directly to 6 after finding that 3^6 = 729, completely ignoring the effect of the square root radical on the left side. Final Answer Therefore, the correct answer is 12.

Question 8

The value of $\left( \frac{9^2 	imes 18^4}{3^{16}} ight)$ is

Accepted Answer

Concept & Formula: This problem requires decomposing composite bases into their prime factors to simplify a complex fractional exponential expression. The primary index laws used are: (a × b)^n = a^n × b^n (a^m)^n = a^{m × n} (a^m/a^n) = a^{m - n} Step-by-Step Solution: • Given: The expression to evaluate is (9^2 × 18^4/3^{16)}. • Calculation / Deduction: • Step 1: Prime factorize all the bases in the numerator. • 9 = 3^2 • 18 = 2 × 9 = 2 × 3^2 • Step 2: Substitute these prime factorizations back into the expression. ((3^2)^2 × (2 × 3^2)^4/3^{16)} • Step 3: Apply the power rules to remove the brackets. • (3^2)^2 = 3^4 • (2 × 3^2)^4 = 2^4 × (3^2)^4 = 2^4 × 3^8 • The numerator becomes: 3^4 × 2^4 × 3^8 • Step 4: Combine the terms with the same base in the numerator. • 2^4 × (3^4 × 3^8) = 2^4 × 3^{12} • Step 5: Substitute the simplified numerator back over the denominator. \frac{2^4 × 3^{12}}{3^{16}} • Step 6: Apply the division rule of indices for base 3. 2^4 × 3^{12 - 16} = 2^4 × 3^{-4} • Step 7: Convert the negative exponent into a fraction and evaluate. (2^4/3^4) = (16/81) Exam Strategy & Shortcut: Break numbers down to prime bases immediately. Once you see 9 and 18, you know the entire expression consists of base 2 and base 3. Tally the powers of 3 in the numerator mentally: 4 (from 9^2) + 8 (from 18^4) = 12. Tally powers of 3 in denominator: 16. Net power of 3 is -4 (meaning 3^4 or 81 is in the denominator). Tally powers of 2 in numerator: 4 (from 18^4). Numerator is 2^4 = 16. The answer is (16/81). Common Pitfall: A standard error is miscalculating the expansion of (2 × 3^2)^4. Some students forget to distribute the power of 4 to the 2, resulting in an incorrect factor of 2 instead of 16. Always distribute the outer exponent to every single factor inside the parenthesis. Final Answer: Therefore, the correct answer is (16/81).

Question 9

$(19)^{12} 	imes (19)^8 \div (19)^4 = (19)^x$

Accepted Answer

Concept & Formula: This problem tests the direct application of the product and quotient rules of indices for a uniform base. Since the base is identical across all terms, we only need to perform basic arithmetic on the exponents. a^m × a^n = a^{m + n} (a^m/a^n) = a^{m - n} Step-by-Step Solution: • Given: The equation is (19)^{12} × (19)^8 \div (19)^4 = (19)^x. • Calculation / Deduction: • Since all bases are exactly the same (19), we can combine the operations into a single exponent equation. • For multiplication, add the exponents: 12 + 8. • For division, subtract the exponent: - 4. • Combine them on the left side: (19)^{12 + 8 - 4} = (19)^x (19)^{20 - 4} = (19)^x (19)^{16} = (19)^x • By equating the powers of the identical bases, we find: x = 16 • Reviewing the provided options: (a) 6, (b) 8, (c) 12, (d) 24. The correct answer 16 is not listed. Exam Strategy & Shortcut: You do not need to rewrite the bases. Simply isolate the exponents visually and calculate mentally: 12 + 8 = 20. Then 20 - 4 = 16. Scan the options for 16. If it is absent, immediately confidently select "None of these". This should take roughly 3 seconds. Common Pitfall: A common mistake is misinterpreting the order of operations or the exponent rules, such as multiplying the exponents instead of adding them, or getting confused and applying division to the base numbers themselves. Stick strictly to the index laws. Final Answer: Therefore, the correct answer is None of these.

Question 10

$(3)^8 	imes (3)^4 = x$

Accepted Answer

Concept & Formula: This question requires applying the product rule of indices first, and then converting the resulting base into a larger, equivalent exponential form to match the options. a^m × a^n = a^{m + n} a^{m × n} = (a^m)^n Step-by-Step Solution: • Given: The expression is (3)^8 × (3)^4. • Calculation / Deduction: • First, apply the product rule since the bases are identical: 3^{8 + 4} = 3^{12} • Our simplified result is 3^{12}. Now we must test the options to find an equivalent value. • Let's test the base 27 options: We know 27 = 3^3. • Option (a): (27)^3 = (3^3)^3 = 3^9. (Incorrect) • Option (b): (27)^5 = (3^3)^5 = 3^{15}. (Incorrect) • Let's test the base 729 options: We know 729 = 3^6. • Option (c): (729)^2 = (3^6)^2 = 3^{12}. (Matches exactly!) • Option (d): (729)^3 = (3^6)^3 = 3^{18}. (Incorrect) Exam Strategy & Shortcut: Once you calculate 3^{12}, quickly scan the bases in the options: 27 and 729. Recognize that 27 = 3^3 and 729 = 3^6. To get a power of 12, you need either (3^3)^4 or (3^6)^2. The option (729)^2 perfectly represents (3^6)^2. Common Pitfall: A very common mistake is finding 3^{12}, not seeing it immediately in the options, and impulsively selecting "None of these". Exam creators intentionally format the correct answer with a different base to ensure you truly understand how exponent manipulation works in both directions. Final Answer: Therefore, the correct answer is (729)^2.

Question 11

$\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}} = $x$

Accepted Answer

Concept & Formula: This question requires solving a finite series of nested square roots of the same number. There is a direct algebraic formula to resolve these specific structures instantly. \sqrt{a\sqrt{a\sqrt{a...n times}}} = a^{(2^n-1/2^n)} Step-by-Step Solution: • Given: The expression is \sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}. • Calculation: 1. Identify the repeating base value (a) and the total number of square roots (n). a = 2 n = 5 2. Calculate the denominator for the fractional exponent, which is 2^n: 2^5 = 32 3. Calculate the numerator for the fractional exponent, which is 2^n - 1: 32 - 1 = 31 4. Substitute these values into the standard formula: 2^{(31/32)} Exam Strategy & Shortcut: Do not attempt to solve this from the inside out manually during an exam. Simply count the number of radical signs. There are 5 roots. Calculate 2^5 = 32. The power of the base number will always be a fraction where the denominator is this result (32), and the numerator is exactly one less (31). This immediately points to option (b). Common Pitfall: A frequent mistake is confusing this finite nested root structure with an infinite nested root structure (\sqrt{2\sqrt{2...}} to infinity). For an infinite series, the answer is simply the base number itself (2). Always check if the series terminates. Final Answer: Therefore, the correct answer is 2^{(31/32)}.

Question 12

$9^3 	imes (81)^2 \div (27)^3 = (3)^x$

Accepted Answer

Concept & Formula: This problem evaluates your ability to standardize bases and apply the core laws of indices to simplify algebraic expressions. a^m × a^n = a^{m+n} a^m \div a^n = a^{m-n} (a^m)^n = a^{mn} Step-by-Step Solution: • Given: The expression to evaluate is 9^3 × (81)^2 \div (27)^3 = (3)^x. • Calculation: 1. Observe that all numbers (9, 81, 27) are powers of the base 3. Convert each term accordingly: 9 = 3^2 81 = 3^4 27 = 3^3 2. Substitute these prime bases back into the original equation: (3^2)^3 × (3^4)^2 \div (3^3)^3 = 3^x 3. Apply the power of a power rule to simplify the exponents: 3^6 × 3^8 \div 3^9 = 3^x 4. Apply the product and quotient rules for exponents from left to right: 3^{(6 + 8 - 9)} = 3^x 3^{14 - 9} = 3^x 3^5 = 3^x 5. Since the bases on both sides are identical, equate the exponents: x = 5 Exam Strategy & Shortcut: Do not write out full base conversions. Mentally translate the problem purely into the powers of base 3. You know 9 is 3^2, so 9^3 gives a power of 6. 81 is 3^4, so squared it gives 8. 27 is 3^3, so cubed it gives 9. Combine them directly: 6 + 8 - 9 = 5. The answer is 5 in seconds. Common Pitfall: Students often confuse the rules when raising a power to another power, mistakenly adding the exponents instead of multiplying them (e.g., computing (3^2)^3 as 3^5 instead of 3^6). Always multiply when brackets separate the powers. Final Answer: Therefore, the correct answer is 5.

Question 13

Which of the following expressions has the greatest value?

Accepted Answer

Concept & Formula: This problem tests your mastery of the power of a power rule and the zero exponent rule. When an exponent is raised to another exponent, you multiply the powers. Any non-zero number raised to the power of zero equals one. (x^a)^b = x^{ab} x^0 = 1 Step-by-Step Solution: • Given: Four separate exponential expressions to evaluate and compare. • Calculation: 1. Evaluate option (a): [(2^{-1})^0]^2 Multiply the exponents: -1 × 0 × 2 = 0. Result: 2^0 = 1. 2. Evaluate option (b): [(4^0)^{-(1/2)}]^2 Multiply the exponents: 0 × -(1/2) × 2 = 0. Result: 4^0 = 1. 3. Evaluate option (c): [(2^{-2})^{-1}]^2 Multiply the exponents: -2 × -1 × 2 = 4. Result: 2^4 = 16. 4. Evaluate option (d): [(2^{-1})^2]^2 Multiply the exponents: -1 × 2 × 2 = -4. Result: 2^{-4} = (1/16). 5. Compare the results: 1, 1, 16, and (1/16). The greatest value is 16. Exam Strategy & Shortcut: Scan the options for the number 0 in the exponent chain. In options (a) and (b), there is a 0 in the multiplication sequence of the powers. Because anything multiplied by 0 is 0, you instantly know those expressions evaluate to a base raised to 0, which is 1. You can immediately eliminate them and only calculate (c) and (d). Recognizing the double negatives in (c) guarantees a positive power, while (d) remains a negative power (a fraction), making (c) the obvious largest choice. Common Pitfall: A common mistake is trying to evaluate the inside of the brackets first instead of just multiplying all the exponents together simultaneously. This wastes time and increases the chance of arithmetic errors, especially with fractions and negative signs. Final Answer: Therefore, the correct answer is [(2^{-2})^{-1}]^2.

Question 14

$(10)^{24} 	imes (10)^{-21} = $x$

Accepted Answer

Concept & Formula: This question tests the fundamental product rule of exponents. When multiplying two terms with the exact same base, you simply add their exponents together. a^m × a^n = a^{m+n} Step-by-Step Solution: • Given: The expression to simplify is (10)^{24} × (10)^{-21}. • Calculation: 1. Confirm that both terms share the same base, which is 10. 2. Apply the product rule by adding the exponents: 10^{(24 + (-21))} 3. Simplify the exponent arithmetic: 10^{(24 - 21)} = 10^3 4. Calculate the final standard numerical value: 10 × 10 × 10 = 1000 Exam Strategy & Shortcut: This is a direct application of a foundational rule. The moment you see identical bases being multiplied, immediately look at the powers. Mentally execute 24 - 21 = 3. Knowing 10^3 = 1000 takes only a fraction of a second. No written steps are necessary. Common Pitfall: Students rushing through the exam might mistakenly subtract the exponents incorrectly due to the negative sign, perhaps computing 24 - (-21) = 45, or they might multiply the bases to get 100 and then do something with the powers. Always remember: keep the base, add the powers. Final Answer: Therefore, the correct answer is 1000.

Question 15

The value of $(256)^{\frac{5}{4}}$ is

Accepted Answer

Concept & Formula: This problem evaluates your ability to work with fractional exponents. The denominator of a fractional exponent represents the root, and the numerator represents the power. Finding a prime base is the most efficient way to solve this. x^{(m/n)} = (\sqrt[n]{x})^m = (x^{(1/n)})^m Step-by-Step Solution: • Given: The expression to evaluate is (256)^{(5/4)}. • Calculation: 1. Identify the base (256) and express it as a smaller exponent to easily interact with the fractional power. We know 256 = 4^4. (Alternatively, 2^8). Let us use 4^4 as it perfectly cancels the denominator. 2. Substitute this into the expression: (4^4)^{(5/4)} 3. Apply the power of a power rule (x^a)^b = x^{ab} by multiplying the exponents: 4 × (5/4) = 5 4. The expression simplifies to: 4^5 5. Calculate the final power: 4^5 = 4^4 × 4 = 256 × 4 = 1024 Exam Strategy & Shortcut: Look at the denominator of the fractional exponent—it is 4. This is a major clue that 256 is something to the power of 4. Recognizing 256 as 4^4 instantly allows you to cancel the 4s, leaving you with 4^5. If you know your powers of 2, 2^{10} is a famous benchmark number in computer science (1 kilobyte), which is 1024. Common Pitfall: A standard pitfall is trying to calculate the 4th root of 256 without converting it to a base exponent first. While finding \sqrt[4]{256} = 4 is doable, students often get stuck on larger numbers if they do not instinctively break them down into prime or smaller bases. Final Answer: Therefore, the correct answer is 1024.

Question 16

The value of $(\sqrt{8})^{\frac{1}{3}}$ is

Accepted Answer

Concept & Formula: This question requires converting a radical (square root) into a fractional exponent, and then applying the power of a power rule. \sqrt[n]{x} = x^{(1/n)} (x^a)^b = x^{ab} Step-by-Step Solution: • Given: The expression to evaluate is (\sqrt{8})^{(1/3)}. • Calculation: 1. Convert the square root of 8 into a fractional exponent: \sqrt{8} = 8^{(1/2)} 2. Substitute this back into the original expression: (8^{(1/2)})^{(1/3)} 3. Apply the power of a power rule by multiplying the fractions: 8^{((1/2) × (1/3))} = 8^{(1/6)} 4. Convert the base 8 into a prime base to simplify further. We know 8 = 2^3. (2^3)^{(1/6)} 5. Multiply the exponents again: 2^{(3 × (1/6))} = 2^{(3/6)} 6. Simplify the fraction in the exponent: 2^{(1/2)} 7. Convert the fractional exponent back to a radical for the final answer: \sqrt{2} Exam Strategy & Shortcut: Reverse the order of operations for speed. The power (1/3) is a cube root. The expression is effectively the cube root of the square root of 8. Because 8 is a perfect cube (2^3), handle the cube root first! The cube root of 8 is 2. You are left with just the square root over that 2, resulting instantly in \sqrt{2}. Common Pitfall: Students often struggle with nested radicals and fractional powers. A common mistake is to multiply the base by the exponent (e.g., 8 × (1/3)) instead of recognizing it as a root. Always convert roots to fractions if you are unsure how to proceed. Final Answer: Therefore, the correct answer is \sqrt{2}.

Question 17

Simplified form of $[(\sqrt[5]{x^{-\frac{3}{5}}})^{-\frac{5}{3}}]^5$ is

Accepted Answer

Concept & Formula: This problem tests the chaining of power rules and radical rules. Instead of working layer by layer, you can convert all radicals into fractional exponents and multiply the entire chain of exponents together. \sqrt[n]{x} = x^{(1/n)} (((x^a)^b)^c)^d = x^{a × b × c × d} Step-by-Step Solution: • Given: The nested expression is [(\sqrt[5]{x^{-(3/5)}})^{-(5/3)}]^5. • Calculation: 1. First, convert the 5th root into a fractional exponent to standardize all operations. The 5th root becomes an exponent of (1/5). 2. Rewrite the entire expression using only exponents: (((x^{-(3/5)})^{(1/5)})^{-(5/3)})^5 3. Since these are all nested powers, apply the power of a power rule by multiplying the entire chain of exponents together: Exponent = -(3/5) × (1/5) × -(5/3) × 5 4. Notice the negative signs cancel out (negative × negative = positive): Exponent = (3/5) × (1/5) × (5/3) × 5 5. Simplify the fractions. The 3s cancel, and the two 5s in the numerator cancel the two 5s in the denominator: Exponent = 1 6. The final simplified base and exponent is: x^1 = x Exam Strategy & Shortcut: Avoid rewriting the variable x over and over. Just extract the exponents and multiply them linearly. You have -3/5. It is under a 5th root, so multiply by 1/5. It is raised to -5/3, so multiply by that. Finally, it is raised to 5, so multiply by that. -(3/5) × (1/5) × -(5/3) × 5. Visually cross out the corresponding numerators and denominators. Everything cancels out perfectly leaving a 1. The answer is just x. Common Pitfall: Students often get confused by the sequence and try to evaluate the innermost terms fully before moving outward, which leads to messy fractions like x^{-3/25} and increases the likelihood of a calculation error. Stringing them together as a single multiplication problem is much safer. Final Answer: Therefore, the correct answer is x.

Question 18

The value of $$ \frac{1}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{-\frac{3}{4}}} + \frac{1}{(32)^{-\frac{1}{5}}} $$ is

Accepted Answer

Concept & Formula The problem tests the laws of negative exponents and fractional indices. a^{-m} = (1/a^m) \implies (1/a^{-m)} = a^m (x^a)^b = x^{ab} Step-by-Step Solution • First term: (1/(216)^{-\frac{2){3}}} = (216)^{(2/3)} . Since 216 = 6^3 , we apply the power rule to get (6^3)^{(2/3)} = 6^2 = 36 . • Second term: (1/(256)^{-\frac{3){4}}} = (256)^{(3/4)} . Since 256 = 4^4 , we get (4^4)^{(3/4)} = 4^3 = 64 . • Third term: (1/(32)^{-\frac{1){5}}} = (32)^{(1/5)} . Since 32 = 2^5 , we get (2^5)^{(1/5)} = 2^1 = 2 . • Add the simplified terms together: 36 + 64 + 2 = 102 . Exam Strategy & Shortcut Recognize perfect powers immediately. The numbers 216, 256, and 32 are common powers ( 6^3 , 4^4 , and 2^5 ). The denominators of the given fractional exponents (3, 4, and 5) match these powers perfectly, meaning the bases will simplify to integers. A quick mental calculation of 6^2 + 4^3 + 2^1 yields 102 in seconds without writing anything down. Common Pitfall Students often mishandle the double negative when moving a term from the denominator to the numerator, mistakenly keeping the fractional exponent negative and getting stuck with complex fractions. Remember that flipping the position vertically strictly flips the exponent's sign. Final Answer Therefore, the correct answer is 102.

Question 19

$$ (2.4 	imes 10^3) \div (8 	imes 10^{-2}) = $$ $x$

Accepted Answer

Concept & Formula This question requires combining scientific notation arithmetic and exponent rules for division. (a × 10^m/b × 10^n) = ((a/b)) × 10^{m-n} Step-by-Step Solution • Group the numerical coefficients and the base-10 powers separately: ((2.4/8)) × ((10^3/10^{-2)}) • Divide the coefficients. Since (24/8) = 3 , it follows that (2.4/8) = 0.3 . • Subtract the exponent of the denominator from the numerator's exponent: 3 - (-2) = 3 + 2 = 5 • Combine them to form 0.3 × 10^5 . • Convert to standard scientific form by moving the decimal point one place to the right, which decreases the exponent by 1: 0.3 × 10^5 = 3 × 10^4 Exam Strategy & Shortcut Treat 2.4 as 24 and divide by 8 to immediately get 3. This tells you the coefficient is related to 3. Then, quickly calculate the power of 10: 3 - (-2) = 5 . Since we shifted a decimal (0.3 instead of 3), 10^5 becomes 10^4 . You can mentally verify the order of magnitude to lock in the correct option in under 15 seconds. Common Pitfall Students often mishandle the subtraction of a negative exponent, mistakenly calculating 3 - 2 = 1 instead of 3 - (-2) = 5 . Always write out or mentally vocalize the double negative explicitly when dividing exponents with negative signs. Final Answer Therefore, the correct answer is 3 × 10^4 .

Question 20

If $$ \left(\frac{3}{5}\right)^3 \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{2x-1} $$, then $x$ is equal to

Accepted Answer

Concept & Formula This problem tests the product rule of exponents and the property of equating exponents when the bases on both sides of an equation are identical. a^m × a^n = a^{m+n} If a^m = a^n , then m = n (given a 
eq 0, 1, -1 ). Step-by-Step Solution • Observe the left-hand side of the equation. Both terms have the same base: (3/5) . • Apply the product rule by adding the exponents: ((3/5))^{3 + (-6)} = ((3/5))^{-3} • Now, rewrite the full equation with the simplified left side: ((3/5))^{-3} = ((3/5))^{2x-1} • Since the bases are exactly the same on both sides, we can drop the bases and equate the exponents directly: -3 = 2x - 1 • Solve the linear equation for x. Add 1 to both sides: -3 + 1 = 2x -2 = 2x • Divide by 2: x = -1 Exam Strategy & Shortcut Since the bases across the entire equation are identical (3/5), you can bypass writing them down entirely. Jump straight to the exponent arithmetic: 3 - 6 = 2x - 1. That simplifies mentally to -3 = 2x - 1, meaning 2x = -2, so x = -1. This takes less than 5 seconds without writing a single fraction. Common Pitfall A common error is mishandling the negative numbers during basic addition and subtraction. Students sometimes calculate 3 - 6 = 3 instead of -3, which leads to the incorrect equation 3 = 2x - 1, resulting in x = 2. Pay strict attention to signs during exponent arithmetic. Final Answer Therefore, the correct answer is -1.

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