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

$5^0 	imes 8 = x$

Accepted Answer

Concept & Formula: This problem tests the Zero Exponent Rule. According to the laws of indices, any non-zero number raised to the power of zero is always equal to 1. a^0 = 1 (where a 
eq 0) Step-by-Step Solution: • Given: The expression to evaluate is 5^0 × 8. • Calculation / Deduction: • First, simplify the exponential term using the Zero Exponent Rule. • 5^0 = 1 • Substitute this value back into the original expression. • 1 × 8 = 8 Exam Strategy & Shortcut: Evaluate 5^0 as 1 instantly in your head. Do not waste time writing down intermediate steps. The moment you see a zero power on a non-zero base, treat it as a multiplication by 1. Common Pitfall: A very common error is assuming that any number raised to the power of zero equals 0 (i.e., thinking 5^0 = 0). If you make this mistake, your next step will be 0 × 8 = 0, leading you straight to the incorrect option (a). Always remember the zero exponent equals one. Final Answer: Therefore, the correct answer is 8.

Question 9

If $289 = 17^{\frac{1}{5}x}$, then $x$ is

Accepted Answer

Concept & Formula: This problem requires solving an exponential equation by equating the bases. If two exponential terms with the same base are equal, their exponents must also be equal. a^m = a^n \implies m = n Step-by-Step Solution: • Given: The equation is 289 = 17^{(1/5)x}. • Calculation / Deduction: • First, express the left-hand side (289) in terms of the base on the right-hand side (17). • We know that 17 × 17 = 289, so 289 = 17^2. • Substitute this back into the equation: 17^2 = 17^{(x/5)} • Since the bases (17) are now identical on both sides, we can equate the exponents: 2 = (x/5) • Multiply both sides by 5 to isolate x: x = 2 × 5 x = 10 • Looking at the given options: (a) (2/5), (b) 8, (c) 16, (d) 32. Since 10 is not listed, the answer is "None of these". Exam Strategy & Shortcut: Memorize squares up to 30. Recognizing immediately that 289 is 17^2 allows you to skip writing the base and directly solve the exponent equation 2 = (x/5) \implies x = 10 in your head in under 10 seconds. Common Pitfall: A common error is mishandling the fractional exponent during algebraic manipulation. A student might incorrectly divide by 5 instead of multiplying, arriving at x = (2/5), which is a trap answer provided in option (a). Final Answer: Therefore, the correct answer is None of these.

Question 10

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 11

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

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 12

$(64)^4 \div (8)^5 = x$

Accepted Answer

Concept & Strategy: To divide exponential terms with different bases, you must express the larger base in terms of the smaller base if possible. Once the bases are matched, apply the quotient rule of indices. (a^m)^n = a^{m × n} (a^m/a^n) = a^{m - n} Step-by-Step Solution: • Given: The expression is (64)^4 \div (8)^5. • Calculation / Deduction: • We can observe that the larger base (64) is a perfect square of the smaller base (8). • Express 64 as a power of 8: 64 = 8^2 • Substitute this back into the original expression: (8^2)^4 \div 8^5 • Apply the power of a power rule to the first term: 8^{2 × 4} \div 8^5 8^8 \div 8^5 • Now, apply the quotient rule of indices: 8^{8 - 5} = 8^3 • Look at the given options: (a) (8)^8, (b) (8)^2, (c) (8)^{12}, (d) (8)^4. • The calculated answer 8^3 is not present in the options. Exam Strategy & Shortcut: Use mental math. Instantly recognize 64 as 8^2. Mentally multiply the exponent 2 × 4 to get 8. Then subtract the dividing exponent 5. 8 - 5 = 3. The answer is 8^3. Since it's not an option, pick (e) immediately without putting pen to paper. Common Pitfall: A classic mistake is dividing the bases directly without properly resolving the exponents, falsely assuming 64 \div 8 = 8 and then guessing the exponent. Another error is doubting your own correct calculation of 8^3 simply because it isn't among the first four options. Trust your math. Final Answer: Therefore, the correct answer is None of these.

Question 13

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

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 14

$\frac{343 	imes 49}{216 	imes 16 	imes 81} = x$

Accepted Answer

Concept & Strategy: When dealing with a massive fraction of composite numbers, immediately convert every number into its prime or fundamental exponential base. This transforms a difficult arithmetic problem into a simple index addition problem. Step-by-Step Solution: • Given: The fraction is (343 × 49/216 × 16 × 81). • Calculation / Deduction: • Step 1: Convert the numbers in the numerator to their base powers. • 343 = 7^3 • 49 = 7^2 • The numerator becomes: 7^3 × 7^2 = 7^{3 + 2} = 7^5 • Step 2: Convert the numbers in the denominator to their base powers. Look for patterns related to base 6, as the options suggest it. • 216 = 6^3 • For the remaining terms, factorize them: • 16 = 2^4 • 81 = 3^4 • Combine 16 × 81 = 2^4 × 3^4. Because the exponents are the same, we can group the bases: (2 × 3)^4 = 6^4. • The denominator becomes: 6^3 × 6^4 = 6^{3 + 4} = 6^7 • Step 3: Combine the simplified numerator and denominator. (7^5/6^7) • Checking the options, this matches Option (a) perfectly. Exam Strategy & Shortcut: Use the options as a cheat sheet. The options show numerators of base 7 and denominators of base 6. This tells you exactly what bases to target. You only need to verify the powers. Numerator: 343 × 49 is obviously 7^3 × 7^2 = 7^5. Denominator: 216 is 6^3. The rest is 16 × 81 = (2 × 3)^4 = 6^4. Total power of 6 is 3+4=7. The answer is instantly (7^5/6^7). Common Pitfall: The worst approach you can take is trying to multiply out the numerator (16807) and denominator (279936) to create a massive fraction, and then attempting to simplify it. Always look at the format of the options before you start calculating; they dictate the most efficient path. Final Answer: Therefore, the correct answer is (7^5/6^7).

Question 15

The value of $(-\frac{1}{216})^{-\frac{2}{3}}$ is

Accepted Answer

Concept & Formula: This problem tests the rules of negative and fractional indices. A negative exponent indicates the reciprocal of the base, and a fractional exponent indicates a root and a power. ((a/b))^{-n} = ((b/a))^n x^{(m/n)} = (\sqrt[n]{x})^m Step-by-Step Solution: • Given: The expression to evaluate is (-(1/216))^{-(2/3)}. • Calculation: 1. Apply the negative exponent rule by taking the reciprocal of the base fraction to make the exponent positive: (-216)^{(2/3)} 2. Identify the base as a perfect cube. We know that (-6) × (-6) × (-6) = -216. (-6)^3 = -216 3. Substitute this back into the expression: ((-6)^3)^{(2/3)} 4. Apply the power of a power rule (x^a)^b = x^{ab} by multiplying the exponents: 3 × (2/3) = 2 5. The expression simplifies to: (-6)^2 6. Calculate the final square: 36 Exam Strategy & Shortcut: When you see a denominator of 3 in the power, immediately look for a cube root. The cube root of -216 is -6. Then look at the numerator of the power, which is 2. Squaring -6 gives 36. Because the power is negative, you flip the original fraction (1/216) to 216 first. The sequence of thought is: Flip to 216, cube root to 6, square to 36. Common Pitfall: Students often drop the negative sign inside the parenthesis too early or assume the negative exponent makes the final answer negative. Remember that squaring any real number, including negative bases like -6, results in a positive value. Final Answer: Therefore, the correct answer is 36.

Question 16

$\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 17

What will come in place of both the question marks in the following question?  $\frac{$x$^{\frac{2}{3}}}{42} = \frac{5}{$x$^{\frac{1}{3}}}$

Accepted Answer

Concept & Formula: This problem relies on cross-multiplication for fractional equations and the fundamental rule of multiplying exponents with the same base. a^m × a^n = a^{m+n} Step-by-Step Solution: • Given: The equation is (x^{\frac{2/3)}}{42} = (5/x^{\frac{1){3}}}. • Calculation: 1. Perform cross-multiplication to isolate the variable terms on one side and the constant terms on the other side: x^{(2/3)} × x^{(1/3)} = 42 × 5 2. Apply the product rule for exponents on the left side by adding the powers: (2/3) + (1/3) = (3/3) = 1 3. The left side simplifies to x^1, which is simply x. x = 42 × 5 4. Calculate the final product on the right side: x = 210 Exam Strategy & Shortcut: Train yourself to spot complementary fractions. The powers (2/3) and (1/3) instantly add up to exactly 1. Recognizing this means you do not need to write out intermediate steps. You can immediately see that cross-multiplying will yield x^1 = 42 × 5. You can mentally calculate 40 × 5 = 200 and 2 × 5 = 10, summing to 210. Common Pitfall: Students sometimes get intimidated by fractional powers and attempt to cube both sides of the equation prematurely. While cubing both sides works, it creates unnecessarily large numbers (42^3 and 5^3) that waste significant time and increase the risk of arithmetic errors. Final Answer: Therefore, the correct answer is 210.

Question 18

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

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 19

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 20

$(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.

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