Question 1

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 2

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 3

What number should be divided by $\sqrt{0.25}$ to give the result as 25?

Accepted Answer

Concept & Logic Translate the word problem directly into an algebraic equation, resolve the square root of the decimal first, and then isolate the unknown variable. Step-by-Step Solution • Given: Let the unknown number be x. The problem states: (x/\sqrt{0.25)} = 25. • Calculation: First, evaluate the square root in the denominator. \sqrt{0.25} = 0.5 • Substitute this back into the equation: (x/0.5) = 25 • Multiply both sides by 0.5 to isolate x: x = 25 × 0.5 • Dividing by 2 is the same as multiplying by 0.5: x = 12.5 Exam Strategy & Shortcut Memorize basic decimal roots. \sqrt{0.25} is instantly 0.5 (or 1/2). The question effectively asks: "What number divided by half equals 25?". Dividing by a half is the same as multiplying by 2. So, x × 2 = 25. Therefore, x must be 12.5. Common Pitfall Incorrectly evaluating \sqrt{0.25} as 0.05 instead of 0.5, which throws off the decimal placement of the final answer entirely. Final Answer Therefore, the correct answer is 12.5.

Question 4

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 5

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

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 6

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

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 7

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.

Question 8

If $\sqrt{(x-1)(y+2)} = 7$, $x$ and $y$ being positive whole numbers, then the values of $x$ and $y$ respectively are

Accepted Answer

Concept & Logic Squaring both sides of a radical equation transforms it into a standard algebraic equation. By evaluating the integer factors of the resulting constant, we can easily test the given whole-number options without complex algebraic isolation. Step-by-Step Solution • Given equation: \sqrt{(x-1)(y+2)} = 7 • Square both sides to eliminate the square root: (x-1)(y+2) = 49 • Since x and y are positive whole numbers, the terms (x-1) and (y+2) must be positive integer pairs that multiply to 49. The prime factorization of 49 is 7 × 7 (or 1 × 49). • Test Option (a) where x = 8 and y = 5: • Substitute the values into the factors: (8 - 1)(5 + 2) = (7)(7) = 49. • This perfectly matches our equation, making it the correct pair. Exam Strategy & Shortcut For equations with multiple variables and explicitly given options, Option Elimination (Back-solving) is the fastest approach. Simply plug the pairs (x, y) from the options into the original equation \sqrt{(x-1)(y+2)} = 7. Testing (a) yields \sqrt{(8-1)(5+2)} = \sqrt{7 × 7} = \sqrt{49} = 7. The problem is solved in seconds. Common Pitfall Students often try to algebraically isolate x or y (e.g., writing x = (49/y+2) + 1), which wastes time and complicates the process when the variables are restricted to whole numbers and options are readily provided to test. Final Answer Therefore, the correct answer is 8, 5.

Question 9

If $\sqrt{.04 	imes .4 	imes a} = .004 	imes .4 	imes \sqrt{b}$, then $\frac{a}{b}$ is

Accepted Answer

Concept & Formula To find the ratio of variables trapped under and outside radicals, isolate the variables by squaring both sides of the equation. Converting small decimals to scientific notation significantly reduces arithmetic errors. Step-by-Step Solution • Given: \sqrt{0.04 × 0.4 × a} = 0.004 × 0.4 × \sqrt{b} • Square both sides to remove the square roots: 0.04 × 0.4 × a = (0.004 × 0.4)^2 × b • Convert the decimals to powers of 10 for structured calculation: Left Side: (4 × 10^{-2}) × (4 × 10^{-1}) × a = 16 × 10^{-3} × a • Right Side: ((4 × 10^{-3}) × (4 × 10^{-1}))^2 × b = (16 × 10^{-4})^2 × b = 256 × 10^{-8} × b • Equate and rearrange to find the ratio (a/b): 16 × 10^{-3} × a = 256 × 10^{-8} × b • (a/b) = \frac{256 × 10^{-8}}{16 × 10^{-3}} • Simplify the fraction and exponents: (256/16) × 10^{-8 - (-3)} = 16 × 10^{-5} Exam Strategy & Shortcut Use decimal digit counting. The left side has a decimal with 3 total decimal places under a root. Squaring the right side (which has 3 + 1 = 4 decimal places) results in 8 decimal places. When dividing the right side's factor by the left side's factor to find (a/b), you are essentially doing 10^{-8} \div 10^{-3} = 10^{-5}. The integer coefficient is 16^2 \div 16 = 16. Thus, the answer is instantly 16 × 10^{-5}. Common Pitfall Attempting to multiply the decimals directly (e.g., 0.004 × 0.4 = 0.0016) and then squaring them conventionally (0.00000256) usually leads to miscounting the zeros. Always use scientific notation for operations on very small decimals. Final Answer Therefore, the correct answer is 16 × 10^{-5}.

Question 10

$\sqrt{\frac{48.4}{0.289}}$ is equal to

Accepted Answer

Concept & Math Similar to evaluating other decimal fractions under roots, equalize the number of decimal places before removing the decimal point. Then extract the whole-number roots and convert the resulting improper fraction to a mixed number. Step-by-Step Solution • Given expression: \sqrt{(48.4/0.289)} • The numerator has 1 decimal place, and the denominator has 3. Equalize them by adding two trailing zeros to the numerator: \sqrt{(48.400/0.289)} • Once equalized, drop the decimal points entirely: \sqrt{(48400/289)} • Recognize the core perfect squares: 484 = 22^2 and 289 = 17^2. • Evaluate the root: \frac{\sqrt{484 × 100}}{\sqrt{289}} = (22 × 10/17) = (220/17) • Convert the improper fraction (220/17) into a mixed fraction. Divide 220 by 17: 17 × 10 = 170, and 17 × 2 = 34, so 17 × 12 = 204. • The remainder is 220 - 204 = 16. • This translates to the mixed fraction 12 (16/17). Exam Strategy & Shortcut You can quickly estimate the final division: 220 \div 17. Since 17 × 10 = 170, the quotient is clearly greater than 10 (eliminating Option a). Because 17 × 13 = 221, the quotient must be just under 13. This logic points directly to 12 (16/17) without having to perform long division mechanically. Common Pitfall A frequent error is dropping the decimal point prematurely without padding, writing \sqrt{(484/289)} = (22/17). This directly leads to the incorrect mixed fraction 1 (5/17) (Option a). Always equalize the decimal places first. Final Answer Therefore, the correct answer is 12 (16/17).

Question 11

If $\sqrt{(x-1)(y+2)} = 7$, $x$ and $y$ being positive whole numbers, then the values of $x$ and $y$ respectively are

Accepted Answer

Concept & Logic Squaring both sides of a radical equation transforms it into a standard algebraic equation. By evaluating the integer factors of the resulting constant, we can easily test the given whole-number options without complex algebraic isolation. Step-by-Step Solution • Given equation: \sqrt{(x-1)(y+2)} = 7 • Square both sides to eliminate the square root: (x-1)(y+2) = 49 • Since x and y are positive whole numbers, the terms (x-1) and (y+2) must be positive integer pairs that multiply to 49. The prime factorization of 49 is 7 × 7 (or 1 × 49). • Test Option (a) where x = 8 and y = 5: • Substitute the values into the factors: (8 - 1)(5 + 2) = (7)(7) = 49. • This perfectly matches our equation, making it the correct pair. Exam Strategy & Shortcut For equations with multiple variables and explicitly given options, Option Elimination (Back-solving) is the fastest approach. Simply plug the pairs (x, y) from the options into the original equation \sqrt{(x-1)(y+2)} = 7. Testing (a) yields \sqrt{(8-1)(5+2)} = \sqrt{7 × 7} = \sqrt{49} = 7. The problem is solved in seconds. Common Pitfall Students often try to algebraically isolate x or y (e.g., writing x = (49/y+2) + 1), which wastes time and complicates the process when the variables are restricted to whole numbers and options are readily provided to test. Final Answer Therefore, the correct answer is 8, 5.

Question 12

$\sqrt{\frac{48.4}{0.289}}$ is equal to

Accepted Answer

Concept & Math Similar to evaluating other decimal fractions under roots, equalize the number of decimal places before removing the decimal point. Then extract the whole-number roots and convert the resulting improper fraction to a mixed number. Step-by-Step Solution • Given expression: \sqrt{(48.4/0.289)} • The numerator has 1 decimal place, and the denominator has 3. Equalize them by adding two trailing zeros to the numerator: \sqrt{(48.400/0.289)} • Once equalized, drop the decimal points entirely: \sqrt{(48400/289)} • Recognize the core perfect squares: 484 = 22^2 and 289 = 17^2. • Evaluate the root: \frac{\sqrt{484 × 100}}{\sqrt{289}} = (22 × 10/17) = (220/17) • Convert the improper fraction (220/17) into a mixed fraction. Divide 220 by 17: 17 × 10 = 170, and 17 × 2 = 34, so 17 × 12 = 204. • The remainder is 220 - 204 = 16. • This translates to the mixed fraction 12 (16/17). Exam Strategy & Shortcut You can quickly estimate the final division: 220 \div 17. Since 17 × 10 = 170, the quotient is clearly greater than 10 (eliminating Option a). Because 17 × 13 = 221, the quotient must be just under 13. This logic points directly to 12 (16/17) without having to perform long division mechanically. Common Pitfall A frequent error is dropping the decimal point prematurely without padding, writing \sqrt{(484/289)} = (22/17). This directly leads to the incorrect mixed fraction 1 (5/17) (Option a). Always equalize the decimal places first. Final Answer Therefore, the correct answer is 12 (16/17).

Question 13

If $\sqrt{1 + \frac{55}{729}} = 1 + \frac{x}{27}$, then the value of $x$ is

Accepted Answer

Concept & Formula To solve equations with variables outside a radical, first simplify the numeric expression under the square root, compute its exact root, and then algebraically isolate the unknown variable. Step-by-Step Solution • Given equation: \sqrt{1 + (55/729)} = 1 + (x/27) • Simplify the expression under the square root by finding a common denominator: 1 + (55/729) = (729 + 55/729) = (784/729) • The equation becomes: \sqrt{(784/729)} = 1 + (x/27) • Extract the square root of the numerator and the denominator. We know that 27^2 = 729. For 784, since 25^2 = 625 and 30^2 = 900, and it ends in 4, the root is 28. \frac{\sqrt{784}}{\sqrt{729}} = (28/27) • Substitute this back into the equation: (28/27) = 1 + (x/27) • Subtract 1 from both sides: (28/27) - 1 = (x/27) • (28 - 27/27) = (x/27) • (1/27) = (x/27) • Since the denominators are equal, the numerators must be equal: x = 1 Exam Strategy & Shortcut Recognize the pattern \sqrt{1 + (k/d^2)} = (n/d). Here, the denominator is d = 27 (27^2 = 729). The numerator under the root becomes 729 + 55 = 784. The root is (28/27). The right side is structured as 1 + (x/27) = (27 + x/27). Equating numerators: 28 = 27 + x, which instantly gives x = 1. Common Pitfall Students often try to square both sides of the initial equation right away, resulting in 1 + (55/729) = (1 + (x/27))^2. Expanding the right side creates a quadratic equation (1 + (2x/27) + (x^2/729)), making the math unnecessarily tedious and increasing the risk of calculation errors. Final Answer Therefore, the correct answer is 1.

Question 14

$\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = x$

Accepted Answer

Concept & Math When subtracting fractional radicals, extract the roots of the numerator and denominator separately where possible. Then, rationalize the denominators or find a common denominator to combine the terms. Step-by-Step Solution • Given expression: \sqrt{(4/3)} - \sqrt{(3/4)} = x • Apply the square root separately to the numerators and denominators: \frac{\sqrt{4}}{\sqrt{3}} - \frac{\sqrt{3}}{\sqrt{4}} • Evaluate the perfect squares (\sqrt{4} = 2): (2/\sqrt{3)} - \frac{\sqrt{3}}{2} • Find a common denominator to subtract the fractions. The common denominator is 2\sqrt{3}. • Multiply the first term by (2/2) and the second term by \frac{\sqrt{3}}{\sqrt{3}}: (2 × 2/2\sqrt{3)} - \frac{\sqrt{3} × \sqrt{3}}{2\sqrt{3}} • Simplify the numerators: (4/2\sqrt{3)} - (3/2\sqrt{3)} • Combine the fractions: (4 - 3/2\sqrt{3)} = (1/2\sqrt{3)} • Thus, x = (1/2\sqrt{3)}. Exam Strategy & Shortcut You can treat \sqrt{(4/3)} and \sqrt{(3/4)} as variables a and (1/a) where a = (2/\sqrt{3)}. The expression is a - (1/a). So, (2/\sqrt{3)} - \frac{\sqrt{3}}{2}. Cross-multiplying gives (4 - 3/2\sqrt{3)} = (1/2\sqrt{3)}. This visual shortcut skips redundant radical writing steps. Common Pitfall A common error is attempting to combine the terms under a single root by subtraction, incorrectly assuming \sqrt{a} - \sqrt{b} = \sqrt{a - b}. This leads to fundamentally flawed logic and wrong answers. Radicals must be evaluated or combined using common denominators. Final Answer Therefore, the correct answer is (1/2\sqrt{3)}.

Question 15

The value of $\sqrt{2}$ upto three places of decimal is

Accepted Answer

Concept & Fact The square roots of small non-perfect square integers are irrational numbers. Their approximate decimal values up to three decimal places are standard mathematical constants that must be memorized for competitive exams. Step-by-Step Solution • The question directly asks for the factual decimal value of \sqrt{2}. • While you can calculate it using the long division method for square roots, competitive exams require rote memorization of this specific value. • The exact value is an infinite, non-repeating decimal: 1.41421356... • Rounding or truncating this to three decimal places yields 1.414. • Comparing this to the given options, 1.414 matches option (d). Exam Strategy & Shortcut There is no calculation shortcut; the strategy is purely memorization. You must commit the following core values to memory to save time in geometry and mensuration problems: \sqrt{2} \approx 1.414 \sqrt{3} \approx 1.732 \sqrt{5} \approx 2.236 Common Pitfall Students who haven't memorized the value might attempt to estimate it by squaring the options (e.g., 1.41^2). While 1.4 × 1.4 = 1.96, attempting to manually multiply 1.412 × 1.412 under time pressure consumes valuable minutes and often leads to arithmetic mistakes. Final Answer Therefore, the correct answer is 1.414.

Question 16

$\frac{\sqrt{24} + \sqrt{216}}{\sqrt{96}} = x$

Accepted Answer

Concept & Formula: When dealing with a fraction containing square roots in both the numerator and denominator, simplify all surds to their base forms first. This often reveals a common factor that can be canceled out. Step-by-Step Solution: • Simplify each term in the fraction by finding the largest perfect square factors: • \sqrt{24} = \sqrt{4 × 6} = 2\sqrt{6} • \sqrt{216} = \sqrt{36 × 6} = 6\sqrt{6} • \sqrt{96} = \sqrt{16 × 6} = 4\sqrt{6} • Substitute these simplified forms back into the original fraction: • x = \frac{2\sqrt{6} + 6\sqrt{6}}{4\sqrt{6}} • Combine the like terms in the numerator: • x = \frac{(2 + 6)\sqrt{6}}{4\sqrt{6}} • x = \frac{8\sqrt{6}}{4\sqrt{6}} • Cancel out the common \sqrt{6} term from the numerator and denominator, and divide the integers: • x = (8/4) = 2 Exam Strategy & Shortcut: Notice that all the numbers inside the square roots (24, 216, 96) share a common factor of 24. \sqrt{216} = \sqrt{9 × 24} = 3\sqrt{24} \sqrt{96} = \sqrt{4 × 24} = 2\sqrt{24} Using this, the expression immediately becomes \frac{\sqrt{24} + 3\sqrt{24}}{2\sqrt{24}} = \frac{4\sqrt{24}}{2\sqrt{24}} = 2. This bypasses breaking them all the way down to prime factors. Common Pitfall: Students sometimes try to cancel terms directly across addition without factoring first (e.g., trying to divide 24 by 96). You must always add or subtract the terms in the numerator to form a single product before canceling with the denominator. Final Answer: Therefore, the correct answer is 2.

Question 17

$\sqrt{50} 	imes \sqrt{98}$ is equal to

Accepted Answer

Concept & Formula: When multiplying square roots, you can combine the multiplicands under a single square root before calculating the final value. \sqrt{a} × \sqrt{b} = \sqrt{a × b} Step-by-Step Solution: • Method 1: Factorization • Simplify each surd by extracting perfect square factors: • \sqrt{50} = \sqrt{25 × 2} = 5\sqrt{2} • \sqrt{98} = \sqrt{49 × 2} = 7\sqrt{2} • Multiply the simplified terms: • (5\sqrt{2}) × (7\sqrt{2}) • = (5 × 7) × (\sqrt{2} × \sqrt{2}) • = 35 × 2 = 70 Exam Strategy & Shortcut: Instead of breaking the numbers down, immediately combine them under one root and look for trailing zeros. \sqrt{50 × 98} = \sqrt{4900} Since 49 = 7^2 and 100 = 10^2, the square root of 4900 is instantly recognizable as 70. This skips the intermediate factorization entirely. Common Pitfall: Students sometimes try to calculate the approximate decimal value of \sqrt{50} (around 7.07) and \sqrt{98} (around 9.89) and multiply them. This wastes significant time and introduces rounding errors that might make you pick a distractor option like 70.25. Always simplify algebraically first. Final Answer: Therefore, the correct answer is 70.

Question 18

Given $\sqrt{2} = 1.414$. The value of  $\sqrt{8} + 2\sqrt{32} - 3\sqrt{128} + 4\sqrt{50}$ is:

Accepted Answer

Concept & Formula: Before substituting a given decimal value, always reduce all radical expressions to their simplest base form. By extracting perfect squares, complex surds will collapse into identical like terms. \sqrt{x^2y} = x\sqrt{y} Step-by-Step Solution: • Simplify each term individually to isolate \sqrt{2}: • \sqrt{8} = \sqrt{4 × 2} = 2\sqrt{2} • 2\sqrt{32} = 2\sqrt{16 × 2} = 2(4\sqrt{2}) = 8\sqrt{2} • 3\sqrt{128} = 3\sqrt{64 × 2} = 3(8\sqrt{2}) = 24\sqrt{2} • 4\sqrt{50} = 4\sqrt{25 × 2} = 4(5\sqrt{2}) = 20\sqrt{2} • Substitute these values back into the expression: • 2\sqrt{2} + 8\sqrt{2} - 24\sqrt{2} + 20\sqrt{2} • Factor out the common \sqrt{2} and combine coefficients: • (2 + 8 - 24 + 20)\sqrt{2} • = (30 - 24)\sqrt{2} • = 6\sqrt{2} • Substitute the given decimal value (\sqrt{2} = 1.414): • 6 × 1.414 = 8.484 Exam Strategy & Shortcut: Look closely at the given clue: \sqrt{2} = 1.414. This is a massive hint that every single term in the equation contains a hidden factor of 2. You can immediately divide 8, 32, 128, and 50 by 2 mentally to find the perfect squares (4, 16, 64, 25). Extract their roots directly (2, 4, 8, 5), multiply by their outer coefficients, and sum them up in one fluid mental step to get 6\sqrt{2}. Common Pitfall: Substituting the value of 1.414 too early is a fatal exam mistake. Attempting to calculate \sqrt{8} as \sqrt{4 × 1.414} or similar bizarre math will completely derail your logic. Always simplify to the very end before converting to decimals. Final Answer: Therefore, the correct answer is 8.484.

Question 19

The approximate value of   $\frac{3\sqrt{12}}{2\sqrt{28}} \div \frac{2\sqrt{21}}{\sqrt{98}}$ is

Accepted Answer

Concept & Formula: When dividing fractions containing surds, flip the second fraction (multiply by its reciprocal) and break down all composite numbers under the roots into prime factors to allow for massive cross-cancellation. (a/b) \div (c/d) = (a/b) × (d/c) Step-by-Step Solution: • Rewrite the division as multiplication by inverting the second fraction: • \frac{3\sqrt{12}}{2\sqrt{28}} × \frac{\sqrt{98}}{2\sqrt{21}} • Simplify all surds by pulling out perfect squares: • \sqrt{12} = 2\sqrt{3} • \sqrt{28} = 2\sqrt{7} • \sqrt{98} = 7\sqrt{2} • \sqrt{21} = \sqrt{3}\sqrt{7} (leave as prime factors for cancellation) • Substitute simplified surds into the expression: • \frac{3(2\sqrt{3})}{2(2\sqrt{7})} × \frac{7\sqrt{2}}{2(\sqrt{3}\sqrt{7})} • = \frac{6\sqrt{3}}{4\sqrt{7}} × \frac{7\sqrt{2}}{2\sqrt{3}\sqrt{7}} • Cancel common terms from numerators and denominators: • The \sqrt{3} in the top left cancels with the \sqrt{3} in the bottom right. • Combine the remaining terms: \frac{6 × 7\sqrt{2}}{4 × 2 × (\sqrt{7} × \sqrt{7})} • Simplify the resulting fraction: • \sqrt{7} × \sqrt{7} = 7. • = \frac{42\sqrt{2}}{8 × 7} = \frac{42\sqrt{2}}{56} • Divide top and bottom by 14: = \frac{3\sqrt{2}}{4} • Calculate the final decimal value using \sqrt{2} \approx 1.414: • (3 × 1.414/4) = (4.242/4) = 1.0605 Exam Strategy & Shortcut: Write everything as single large roots to cancel internal factors before pulling out perfect squares. (3/2) × (1/2) × \sqrt{(12 × 98/28 × 21)} Cancel 7 from 98 and 28: \sqrt{(12 × 14/4 × 21)} Cancel 4 from 12 and 4: \sqrt{(3 × 14/21)} Cancel 3 from 3 and 21: \sqrt{(14/7)} = \sqrt{2}. The whole expression elegantly collapses to (3/4) × \sqrt{2}. Much faster than extracting individual perfect squares! Common Pitfall: Forgetting to flip the second fraction when converting division to multiplication is a classic error. It leads you to multiply numerators with numerators, leaving an impossible tangle of roots that won't cancel out properly. Final Answer: Therefore, the correct answer is 1.0605.

Question 20

$\sqrt{110.25} 	imes \sqrt{0.01} \div \sqrt{0.0025} - \sqrt{420.25}$ equals

Accepted Answer

Concept & Logic: This problem combines square roots of decimals with BODMAS/PEMDAS order of operations. Evaluate all square roots first by treating them as perfect squares divided by powers of 10, then apply standard arithmetic precedence (Division/Multiplication, then Subtraction). Step-by-Step Solution: • Resolve the square roots of the decimal numbers: • \sqrt{110.25} = \sqrt{(11025/100)} = (105/10) = 10.5 • \sqrt{0.01} = 0.1 • \sqrt{0.0025} = 0.05 • \sqrt{420.25} = \sqrt{(42025/100)} = (205/10) = 20.5 • Substitute these values back into the equation: • 10.5 × 0.1 \div 0.05 - 20.5 • Apply BODMAS - Perform Division first: • 0.1 \div 0.05 = (0.10/0.05) = 2 • Perform Multiplication next: • 10.5 × 2 = 21 • Finally, perform Subtraction: • 21 - 20.5 = 0.5 Exam Strategy & Shortcut: To quickly find the square roots of numbers ending in 25 (like 11025 and 42025), use the "number ending in 5" squaring trick in reverse. A number ending in 5 squares to something ending in 25. The preceding digits are formed by n(n+1). For 110, 10 × 11 = 110, so the root is 105. For 420, 20 × 21 = 420, so the root is 205. This instantly gives you 10.5 and 20.5 without tedious manual square root division. Common Pitfall: Misplacing decimal points during the square root phase. For example, incorrectly thinking \sqrt{0.0025} is 0.005 instead of 0.05. Always check your root by squaring it mentally (5 × 5 = 25, need 4 decimal places, so 0.05 × 0.05 = 0.0025). Final Answer: Therefore, the correct answer is 0.50.

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