Question 1

The value of $\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}$ is

Accepted Answer

Concept & Strategy: For nested square root problems, always begin evaluating from the innermost square root and work your way outwards step-by-step. Core logic: Simplify from inside out: \sqrt{a + \sqrt{b}} Step-by-Step Solution: - Step 1: Evaluate the innermost root: \sqrt{225} = 15. - Substitute this back: \sqrt{154 + 15}. - Step 2: Add and evaluate the next root: 154 + 15 = 169. - \sqrt{169} = 13. - Substitute this back: \sqrt{108 + 13}. - Step 3: Add and evaluate the next root: 108 + 13 = 121. - \sqrt{121} = 11. - Substitute this back: \sqrt{25 + 11}. - Step 4: Add and evaluate the next root: 25 + 11 = 36. - \sqrt{36} = 6. - Substitute this back into the final, outermost expression: \sqrt{10 + 6}. - Step 5: Final evaluation: 10 + 6 = 16. - \sqrt{16} = 4. Exam Strategy & Shortcut: Look at the outermost operation: \sqrt{10 + something}. Since the total expression evaluates to an integer (based on the options), the sum under the outermost root must be a perfect square greater than 10. The closest perfect square greater than 10 is 16. If the sum is 16, the root is 4. This skips the entire calculation. Common Pitfall: Trying to add the numbers while they are still under their respective square roots. You cannot add terms across different radical layers. You must resolve each radical completely before adding it to the next term. Final Answer: Therefore, the correct answer is 4.

Question 2

Evaluate $\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}$ .

Accepted Answer

Concept & Strategy: Just like other nested radicals, this problem requires solving from the innermost square root outwards. Pay close attention to the alternating addition and subtraction signs. Core logic: Work from the innermost term outwards. Step-by-Step Solution: - Step 1: Evaluate the innermost root: \sqrt{9} = 3. - Substitute this into the expression: \sqrt{19 - 3}. - Step 2: Subtract and evaluate: 19 - 3 = 16. - \sqrt{16} = 4. - Substitute this back into the next layer: \sqrt{21 + 4}. - Step 3: Add and evaluate: 21 + 4 = 25. - \sqrt{25} = 5. - Substitute this into the outermost layer: \sqrt{41 - 5}. - Step 4: Final evaluation: 41 - 5 = 36. - \sqrt{36} = 6. Exam Strategy & Shortcut: Analyze the outermost layer: \sqrt{41 - something positive}. The result must be the square root of a perfect square strictly less than 41. The nearest perfect square less than 41 is 36. The square root of 36 is 6. Option (c) is 6, making it a highly confident guess without doing the inner math. Common Pitfall: Missing the alternating minus and plus signs. Carelessly adding 19 + 3 instead of subtracting, which completely throws off the chain of perfect squares. Always triple-check the operational signs in nested roots. Final Answer: Therefore, the correct answer is 6.

Question 3

$\sqrt{x} 	imes \sqrt{484} = 1034$

Accepted Answer

Concept & Strategy: Isolate the algebraic term \sqrt{x} by evaluating the known perfect square root term and dividing it through to the right-hand side. Core algebraic identity: If \sqrt{x} × a = b, then \sqrt{x} = (b/a) Step-by-Step Solution: - Step 1: Extract the known square root. - Recognize that \sqrt{484} = 22. - Step 2: Substitute and simplify. - \sqrt{x} × 22 = 1034 - \sqrt{x} = (1034/22) - Divide 1034 by 22: \sqrt{x} = 47 - Step 3: Square both sides to find x. - x = 47^2 - 47^2 = (50 - 3)^2 = 2500 - 300 + 9 = 2209 Exam Strategy & Shortcut: After finding \sqrt{x} = 47, determine 47^2 quickly by referencing 50^2 = 2500. Since 47 is just below 50, its square must be somewhat less than 2500. Furthermore, 7^2 = 49, so the final answer must end in 9. Both 2209 and 2401 fit this, but 2401 is 49^2, making 2209 the correct choice. Common Pitfall: Mistakenly taking the square root of 47 instead of squaring it, or halting calculations once you discover 47 because of an operational mixup. Always remember that removing a square root demands squaring the opposing balance. Final Answer: Therefore, the correct answer is 2209.

Question 4

$\sqrt{11881} 	imes \sqrt{x} = 10137$

Accepted Answer

Concept & Strategy: Extract the large integer root first using estimation limits, isolate \sqrt{x} via basic arithmetic division, and square the outcome. Approximation limits: 100^2 = 10000 \quad and \quad 110^2 = 12100 Step-by-Step Solution: - Step 1: Evaluate \sqrt{11881}. - 11881 lies perfectly between 10000 (100^2) and 12100 (110^2). - Since it ends in 1, the unit digit of its root is either 1 or 9. - Given that 11881 is much closer to 12100, we check 109. - Verification: 109^2 = (110 - 1)^2 = 12100 - 220 + 1 = 11881. Thus, \sqrt{11881} = 109. - Step 2: Isolate the unknown term. - 109 × \sqrt{x} = 10137 - \sqrt{x} = (10137/109) - Computing division: \sqrt{x} = 93 - Step 3: Compute x. - x = 93^2 - 93^2 = (100 - 7)^2 = 10000 - 1400 + 49 = 8649 Exam Strategy & Shortcut: Use last digit patterns for the division. In 109 × \sqrt{x} = 10137, the unit digit 9 multiplied by some number must result in a unit digit of 7. Since 9 × 3 = 27, \sqrt{x} must end in 3. Squaring a value ending in 3 (\dots3^2) guarantees the final value x must end in 9. This filters options to 8649 and 9409. Knowing 90^2 = 8100, 93^2 must be close to 8100, confirming 8649. Common Pitfall: Dividing 10137 by 109 incorrectly due to rushing. Take care with multi-digit divisions by rounding 109 to 110 mentally to check magnitude constraints. Final Answer: Therefore, the correct answer is 8649.

Question 5

The square root of $(272^2 - 128^2)$ is

Accepted Answer

Concept & Formula: Avoid squaring large individual numbers directly. Instead, apply the difference of squares algebraic identity to simplify the internal expression. Difference of squares identity: a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: - Step 1: Set up the targeted expression. - We need to compute: \sqrt{272^2 - 128^2} - Step 2: Apply the algebraic identity. - Let a = 272 and b = 128. - 272^2 - 128^2 = (272 - 128)(272 + 128) - Step 3: Compute values within brackets. - 272 - 128 = 144 - 272 + 128 = 400 - So, the expression becomes \sqrt{144 × 400} - Step 4: Factor out the square roots. - \sqrt{144 × 400} = \sqrt{144} × \sqrt{400} - 12 × 20 = 240 Exam Strategy & Shortcut: Never square 272 and 128 manually. The difference of squares structure turns a messy subtraction problem into a product of two obvious perfect squares (\sqrt{144 × 400}). Taking roots gives 12 × 20 = 240 instantly. Common Pitfall: Mistakenly applying the distributive error \sqrt{a^2 - b^2} = a - b, which results in computing 272 - 128 = 144. Radical distribution over subtraction is completely invalid. Final Answer: Therefore, the correct answer is 240.

Question 6

If $y = 5$, then what is the value of $10y\sqrt{y^3 - y^2}$ ?

Accepted Answer

Concept & Strategy: Perform straight algebraic evaluation by replacing all instances of the variable y with its given value, 5. Factor or resolve internal values under the radical before final multiplication. Core step: Substitute y = 5 into the radical function. Step-by-Step Solution: - Step 1: Substitute y = 5 into the statement. - Expression: 10(5) × \sqrt{5^3 - 5^2} - Step 2: Simplify components outside and inside the radical. - Outside product: 10 × 5 = 50 - Inside powers: 5^3 = 125 and 5^2 = 25 - Subtraction under radical: 125 - 25 = 100 - Step 3: Resolve the remaining term. - Expression is now: 50 × \sqrt{100} - Since \sqrt{100} = 10, we evaluate: 50 × 10 = 500 Exam Strategy & Shortcut: Factor out y^2 inside the radical first to work with smaller values: \sqrt{y^3 - y^2} = \sqrt{y^2(y - 1)} = y\sqrt{y - 1}. Substitute y = 5 into this simplified form: 5\sqrt{5 - 1} = 5\sqrt{4} = 5 × 2 = 10. Multiply by the initial front term 10y = 50, yielding 50 × 10 = 500. Common Pitfall: Mistaking 5^3 as 15 or 5^2 as 10 due to rushing. Always write down exponential expansions carefully if you are prone to simple mental arithmetic slips. Final Answer: Therefore, the correct answer is 500.

Question 7

$\sqrt{110\frac{1}{4}} = x$

Accepted Answer

Concept & Strategy: Convert the mixed fraction into an improper fraction, take the square root of the numerator and denominator independently, and then convert back into decimal format. Fraction conversion identity: \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution: - Step 1: Convert the mixed fraction to an improper fraction. - Multiply the integer by the denominator and add the numerator: 110 × 4 + 1 = 440 + 1 = 441. - The expression becomes: \sqrt{(441/4)} - Step 2: Extract the square roots. - Split the radical: \frac{\sqrt{441}}{\sqrt{4}} - We know \sqrt{441} = 21 and \sqrt{4} = 2. - Result: (21/2) - Step 3: Convert the fraction to a decimal value. - (21/2) = 10.5 Exam Strategy & Shortcut: Alternatively, express the value directly as a decimal from the start: \sqrt{110(1/4)} = \sqrt{110.25}. We know that 10^2 = 100 and 11^2 = 121, so the answer must be between 10 and 11. Since the number ends in .25, its square root must end in .5. This points straight to 10.5. Common Pitfall: Mistakenly picking option (a) 10.25 because the original question had a (1/4) fractional component, assuming that the root splits directly onto the component parts. Never split a radical across individual terms of an addition or mixed fraction format. Final Answer: Therefore, the correct answer is 10.5.

Question 8

$$\sqrt{\frac{25}{81} - \frac{1}{9}} = x$$

Accepted Answer

Concept & Formula: To simplify an expression under a square root involving fractions, find a common denominator to subtract the fractions first, then take the square root of the resulting fraction. \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution: • Given: The expression \sqrt{(25/81) - (1/9)} • Calculation: Find the common denominator for 81 and 9, which is 81. Multiply the numerator and denominator of (1/9) by 9: (1/9) = (9/81) Subtract the fractions inside the radical: (25/81) - (9/81) = (16/81) Take the square root: \sqrt{(16/81)} = \frac{\sqrt{16}}{\sqrt{81}} = (4/9) Exam Strategy & Shortcut: Recognize perfect squares immediately. The numbers 25, 81, 9, 16 are all perfect squares. As soon as you compute the numerator subtraction 25 - 9 = 16, you know the answer must be \sqrt{(16/81)} = (4/9) without writing down intermediate steps. Common Pitfall: • Mistake: Distributing the square root over subtraction, i.e., mistakenly computing \sqrt{(25/81)} - \sqrt{(1/9)} = (5/9) - (1/3) = (2/9). • Correction: Remember that \sqrt{A - B} 
eq \sqrt{A} - \sqrt{B}. Always perform the subtraction inside the radical first. Final Answer: Therefore, the correct answer is 4/9.

Question 9

What percentage of the numbers from 1 to 50 have squares that end in the digit 1?

Accepted Answer

Concept & Formula: A number's square ends in the digit 1 if and only if the number itself ends in either 1 or 9 (since 1^2 = 1 and 9^2 = 81). Percentage = ( \frac{Favorable Outcomes}{Total Outcomes} ) × 100 Step-by-Step Solution: • Given: Total count of numbers from 1 to 50 = 50. • Identification: Numbers between 1 and 50 ending in 1 or 9: • Ending in 1: 1, 11, 21, 31, 41 (5 numbers) • Ending in 9: 9, 19, 29, 39, 49 (5 numbers) Total favorable numbers = 5 + 5 = 10. • Calculation: Percentage = ( (10/50) ) × 100 = 20% Exam Strategy & Shortcut: In every block of 10 consecutive numbers (e.g., 1-10, 11-20), exactly 2 numbers end in 1 or 9. Since 2 out of 10 numbers satisfy the condition, the proportion is always (2/10) = 20%, regardless of whether the range goes up to 10, 20, 50, or 100. Common Pitfall: • Mistake: Forgetting to count the numbers ending in 9, or missing the boundary value like 49, leading to an incorrect count of 5 numbers (10%). • Tip: Always write out the unit possibilities (1^2=1 and 9^2=81) explicitly before counting. Final Answer: Therefore, the correct answer is 20.

Question 10

How many perfect squares lie between 120 and 300?

Accepted Answer

Concept & Logic: To find the number of perfect squares within a range, determine the integers whose squares fall strictly within the lower and upper bounds. Step-by-Step Solution: • Given range: Greater than 120 and less than 300. • Find lower limit: 10^2 = 100 \quad (too small) 11^2 = 121 \quad (greater than 120) So, the first perfect square is 11^2. • Find upper limit: 17^2 = 289 \quad (less than 300) 18^2 = 324 \quad (greater than 300) So, the last perfect square is 17^2. • Count the numbers: The integers are 11, 12, 13, 14, 15, 16, 17. Count = 17 - 11 + 1 = 7 Exam Strategy & Shortcut: Memorize squares up to 30. Knowing instantly that 11^2 = 121 and 17^2 = 289 allows you to find the count using the simple formula: Last Number - First Number + 1. 17 - 11 + 1 = 7 Common Pitfall: • Mistake: Off-by-one errors when subtracting limits (17 - 11 = 6) without adding 1 to include both endpoints. • Tip: Always use Upper - Lower + 1 when counting inclusive continuous integer sequences. Final Answer: Therefore, the correct answer is 7.

Question 11

The smallest natural number which is a perfect square and which ends in 3 identical digits lies between

Accepted Answer

Concept & Logic: Perfect squares have strict rules about their ending digits. A perfect square can only end in 0, 1, 4, 5, 6, or 9. Furthermore, if a perfect square ends in multiple identical digits: • It cannot end in 111, 555, 666, or 999 because of modular arithmetic constraints (e.g., an odd square must be a multiple of 8 plus 1). • It cannot end in an odd number of zeros (e.g., 000). • The only possible non-zero repeating identical triplet at the end of a perfect square is 444. Step-by-Step Solution: • Deduction: The square must end in 444. Since the number ends in 4, its square root must end in either 2 or 8 (since 2^2=4 and 8^2=64). • Calculation: Let's test squares ending in 2 or 8, starting from smaller numbers, to find one that ends in 444. • 12^2 = 144 (Ends in 44, not 444) • 22^2 = 484 • 28^2 = 784 • 32^2 = 1024 • 38^2 = 1444 We found it! 1444 is a perfect square (38^2) and it ends in three identical digits (444). • Conclusion: The number 1444 clearly falls into the range between 1000 and 2000. Exam Strategy & Shortcut: Knowing that 38^2 = 1444 is a tremendous shortcut for higher-level aptitude exams. 1444 is the only four-digit perfect square ending in three identical digits. Recognizing this instantly points you to the 1000-2000 bucket. Common Pitfall: • Mistake: Trying to guess numbers ending in 000 (like 1000 or 4000) and thinking they are perfect squares. • Correction: A perfect square must have an even number of trailing zeros. Three zeros is mathematically impossible for a perfect square. Final Answer: Therefore, the correct answer is 1000 and 2000.

Question 12

$$1.5^2 	imes \sqrt{0.0225} = x$$

Accepted Answer

Concept & Strategy: Solve each part of the expression independently by converting decimals to familiar perfect squares, then multiply the results. Step-by-Step Solution: • Given: The equation 1.5^2 × \sqrt{0.0225} • Calculate Part 1: Square 1.5. Ignore the decimal: 15^2 = 225. Since 1.5 has 1 decimal place, 1.5 × 1.5 will have 2 decimal places. 1.5^2 = 2.25 • Calculate Part 2: Find \sqrt{0.0225}. Ignore the decimal: \sqrt{225} = 15. The number 0.0225 has 4 decimal places, so its root will have 2 decimal places. \sqrt{0.0225} = 0.15 • Final Multiplication: Multiply the results. 2.25 × 0.15 Multiply 225 × 15 = 3375. Count the total decimal places: 2 from 2.25 and 2 from 0.15, giving 4 total places. 0.3375 Exam Strategy & Shortcut: Notice that 1.5^2 and \sqrt{0.0225} are structurally related because 15^2 = 225. Rewrite as fractions for safer arithmetic: ((3/2))^2 × \sqrt{(225/10000)} = (9/4) × (15/100) (135/400) = (33.75/100) = 0.3375 Fractional math often prevents decimal misplacement errors under pressure. Common Pitfall: • Mistake: Misplacing the decimal during the final multiplication (2.25 × 0.15), yielding 0.0375 or 3.375. • Correction: When multiplying decimals, ALWAYS add up the decimal places of all factors. 2 places + 2 places MUST equal 4 places in the final answer. Final Answer: Therefore, the correct answer is 0.3375.

Question 13

$\sqrt{1.5625} = x$

Accepted Answer

Concept & Strategy When evaluating the square root of a decimal number, the number of decimal places in the result is exactly half the number of decimal places in the original perfect square. Step-by-Step Solution • Calculation: Disregard the decimal momentarily and focus on finding the square root of the integer 15625. • Since the number ends in 25, its square root must end in 5. • The remaining part of the number is 156. Find two consecutive integers whose product is 156. We know 12 × 13 = 156. • Thus, the integer part of the square root is 12, making the square root of 15625 exactly 125. • Re-insert the decimal point. The original number (1.5625) has 4 decimal places, so the root must have exactly 2 decimal places. • This gives 1.25. Exam Strategy & Shortcut Check the last digit and the integer range. The number ends in 5, so the root ends in 5. The integer part is 1, meaning the root is slightly greater than 1. Test the boundaries: 1.2^2 = 1.44 and 1.3^2 = 1.69. Since 1.5625 falls between them, the root must be 1.25. Common Pitfall Incorrectly placing the decimal point in the final answer (e.g., answering 12.5 instead of 1.25). Remember that 4 decimal places under the root convert to 2 places outside the root. Final Answer Therefore, the correct answer is 1.25.

Question 14

If $\sqrt{.00000676} = .0026$, the square root of $67,60,000$ is:

Accepted Answer

Concept & Logic The significant digits of a square root remain constant if the original number is multiplied by an even power of 10. You can extract the integer square root and append the appropriate number of zeros. \sqrt{x \cdot 10^{2n}} = \sqrt{x} \cdot 10^n Step-by-Step Solution • Given: \sqrt{.00000676} = .0026. By moving the decimal 8 places to the right (multiplying by 10^8), we can deduce that the square root of the significant digits is \sqrt{676} = 26. • Calculation: We need to evaluate \sqrt{6760000}. • Break the number down into known perfect squares: \sqrt{676 × 10000}. • Apply the square root separately to both parts: \sqrt{676} × \sqrt{10000}. • Substitute the known values: 26 × 100 = 2600. Exam Strategy & Shortcut Count the trailing zeros. The number 6760000 has 4 trailing zeros. Its square root will have exactly half that number of trailing zeros, which is 2. The significant integer part is \sqrt{676} = 26, so simply attach two zeros to get 2600. Common Pitfall Overcomplicating the problem by trying to reverse-engineer the decimal shift from the given equation, rather than just extracting the core significant root (\sqrt{676} = 26) and solving the new number independently. Final Answer Therefore, the correct answer is 2600.

Question 15

Given that $\sqrt{13} = 3.605$ and $\sqrt{130} = 11.40$, find the value of $\sqrt{1.3} + \sqrt{1300} + \sqrt{0.013}$.

Accepted Answer

Concept & Strategy When adjusting square roots by powers of 10, you must shift by even powers of 10 (like 100 or 10000) to properly move values in and out of the square root. Use either \sqrt{13} or \sqrt{130} depending on which provides an even decimal shift. Step-by-Step Solution • Given: \sqrt{13} = 3.605 and \sqrt{130} = 11.40. • Calculation: Manipulate each term into a known value divided or multiplied by 100 or 10000: • \sqrt{1.3} = \sqrt{(130/100)} = \frac{\sqrt{130}}{\sqrt{100}} = (11.40/10) = 1.14 • \sqrt{1300} = \sqrt{13 × 100} = \sqrt{13} × 10 = 3.605 × 10 = 36.05 • \sqrt{0.013} = \sqrt{(130/10000)} = \frac{\sqrt{130}}{\sqrt{10000}} = (11.40/100) = 0.114 • Sum the evaluated terms: 1.14 + 36.05 + 0.114 = 37.304 Exam Strategy & Shortcut Focus heavily on the largest number to estimate. \sqrt{1300} is approximately 36. The other terms (\approx 1.1 and \approx 0.1) add a small fraction. The sum must be around 37.2 or 37.3. Option (d) 37.304 is the only perfectly matching candidate. Common Pitfall Incorrectly converting \sqrt{1.3} using \sqrt{13} instead of \sqrt{130} (e.g., writing it as \sqrt{13/10} which leaves a \sqrt{10} in the denominator, making it unnecessarily complex). Final Answer Therefore, the correct answer is 37.304.

Question 16

If $\frac{52}{x} = \sqrt{\frac{169}{289}}$, the value of $x$ is

Accepted Answer

Concept & Formula Resolve the square roots of the known perfect squares on the right side of the equation, then use cross-multiplication to isolate the unknown variable x. \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution • Given: (52/x) = \sqrt{(169/289)} • Calculation: Recognize that 169 and 289 are perfect squares. • \sqrt{169} = 13 and \sqrt{289} = 17. • Substitute these into the equation: (52/x) = (13/17) • Isolate x using cross-multiplication: x = (52 × 17/13) • Simplify the fraction. Notice that 52 is a multiple of 13 (13 × 4 = 52): x = 4 × 17 • x = 68 Exam Strategy & Shortcut Memorizing squares up to 30 is highly recommended. Immediately convert the RHS to 13/17. Notice the horizontal relationship between the numerators: 13 × 4 = 52. Apply the same multiplier to the denominator: 17 × 4 = 68. Common Pitfall Failing to recognize 289 as the perfect square of 17, leading to unnecessary estimation or getting stuck on the simplification step. Final Answer Therefore, the correct answer is 68.

Question 17

If $\sqrt{18 	imes 14 	imes x} = 84$, then $x$ equals

Accepted Answer

Concept & Logic Square both sides of the radical equation to remove the square root, then isolate the unknown variable by dividing. Avoid full multiplication early on to make simplification easier through cancellation. Step-by-Step Solution • Given: \sqrt{18 × 14 × x} = 84 • Calculation: Square both sides to eliminate the radical: 18 × 14 × x = 84^2 18 × 14 × x = 84 × 84 • Isolate x by dividing both sides by (18 × 14): x = (84 × 84/18 × 14) • Simplify the fraction by cancelling out common factors. First, note that 84 \div 14 = 6: x = (6 × 84/18) • Simplify further. Notice that 18 \div 6 = 3: x = (84/3) • Complete the final division: x = 28 Exam Strategy & Shortcut Keep everything factored. When you see 18 × 14 × x = 84 × 84, immediately scan for multiples. 14 × 6 = 84. That leaves 18 × x = 6 × 84. Divide 18 by 6 to get 3x = 84. Therefore, x = 28. This completely avoids any large arithmetic. Common Pitfall Squaring 84 to get 7056 and multiplying 18 × 14 to get 252, and then attempting to divide 7056 \div 252 manually, which is highly prone to calculation errors and wastes precious time. Final Answer Therefore, the correct answer is 28.

Question 18

$\sqrt{\frac{x}{169}} = \frac{54}{39}$

Accepted Answer

Concept & Formula Separate the numerator and denominator under a radical sign to evaluate known perfect squares immediately. Then use basic cross-multiplication. \sqrt{(a/b)} = \frac{\sqrt{a}}{\sqrt{b}} Step-by-Step Solution • Given: \sqrt{(x/169)} = (54/39) • Calculation: Apply the square root separately to the numerator and denominator: \frac{\sqrt{x}}{\sqrt{169}} = (54/39) • Substitute the known square root (\sqrt{169} = 13): \frac{\sqrt{x}}{13} = (54/39) • Simplify the right side of the equation. Notice that both 54 and 39 are divisible by 3: (54/39) = (18/13) • The equation is now: \frac{\sqrt{x}}{13} = (18/13) • Since the denominators are identical, the numerators must be equal: \sqrt{x} = 18 • Square both sides to find x: x = 18^2 x = 324 Exam Strategy & Shortcut Spotting the relationship between the denominators is key. \sqrt{169} is 13. Look at the RHS fraction: 54/39. Recognize that 39 is exactly 13 × 3. So, reducing 54/39 by a factor of 3 immediately gives 18/13. Since the base 13 matches on both sides, \sqrt{x} = 18 \implies x = 324. Common Pitfall Squaring the entire original equation ((x/169) = (2916/1521)) before simplifying the fractions. This results in incredibly large and cumbersome numbers that invite calculation errors. Final Answer Therefore, the correct answer is 324.

Question 19

$\sqrt{\frac{.0196}{x}} = 0.2$

Accepted Answer

Concept & Formula Square both sides of the equation immediately to remove the radical, then cross-multiply to isolate the unknown variable in the denominator. Step-by-Step Solution • Given: \sqrt{(0.0196/x)} = 0.2 • Calculation: Square both sides of the equation to eliminate the square root: (0.0196/x) = (0.2)^2 • Calculate the square of 0.2: (0.0196/x) = 0.04 • Rearrange the equation to isolate x by swapping x and 0.04: x = (0.0196/0.04) • Shift the decimal point two places to the right on both the numerator and denominator to simplify the division: x = (1.96/4) • Perform the division: x = 0.49 Exam Strategy & Shortcut Alternatively, take the square root of the numerator first. \sqrt{196} is 14. Since it has 4 decimal places (0.0196), the root has 2 (0.14). So, (0.14/\sqrt{x)} = 0.2. Therefore, \sqrt{x} = (0.14/0.2) = 0.7. Squaring both sides yields x = 0.49. This avoids dealing with the tiny 0.04 denominator. Common Pitfall Squaring 0.2 incorrectly as 0.2 or 0.004, which corrupts the final fraction and leads to answers that are off by a factor of 10. Final Answer Therefore, the correct answer is 0.49.

Question 20

Which number can replace both the question marks in the equation $\frac{4 \frac{1}{2}}{x} = \frac{x}{32}$.

Accepted Answer

Concept & Formula The problem requires solving a simple algebraic proportion involving fractions. Cross-multiplication is the key principle: if (a/b) = (c/d), then a × d = b × c. Step-by-Step Solution • Given equation with the unknown replaced by x: (4 \frac{1/2)}{x} = (x/32) • Convert the mixed fraction 4 (1/2) to an improper fraction: (9/2). • The equation becomes: (\frac{9/2)}{x} = (x/32) • Cross-multiply to solve for x: x × x = (9/2) × 32 • Simplify the right side: x^2 = 9 × 16 • x^2 = 144 • Take the square root of both sides: x = 12 Exam Strategy & Shortcut Instead of fully multiplying 9 × 16 = 144, leave it as factors: x^2 = 9 × 16. Apply the square root directly to the perfect squares: x = \sqrt{9} × \sqrt{16} = 3 × 4 = 12. This saves calculation time and avoids arithmetic errors on larger numbers. Common Pitfall Students often miscalculate the mixed fraction conversion, writing 4 (1/2) as (5/2) by adding instead of multiplying. Always remember: (Whole × Denominator) + Numerator. Final Answer Therefore, the correct answer is None of these.

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