0.4 is closest to which value?

0.63. 0.4 = 410 = 2 10 23.162 0.63.

If x + 81 = 324, find x.

81. 81 = 9 and 324 = 18, so x = 18 - 9 = 9, giving x = 81.

Simplify 72 + 800 18.

263. 72 = 6 2, 800 = 20 2 and 18 = 3 2. The expression is 26 23 2 = 263.

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Quantitative Aptitude · Numbers & Simplification

Square Roots and Cube Roots — Questions & Answers

These questions ask you to evaluate or estimate square and cube roots, simplify surds, and rationalise denominators. Prime factorisation and grouping digits in pairs (or triples) are the core techniques.

5 solved questions
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Step-by-step AI walkthroughs
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Key formulas included

Key formulas

$\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ and $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$
$\sqrt[3]{a^3 b} = a\sqrt[3]{b}$
Rationalise: $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$ , and $\dfrac{1}{\sqrt{a}+\sqrt{b}} = \dfrac{\sqrt{a}-\sqrt{b}}{a-b}$

5 solved Square Roots and Cube Roots questions

Try each question first — tap an option to check your answer, or open Learn step by step for a guided walkthrough.

01
Evaluate $\sqrt{1225}$ .
5 steps · reveal one at a time
1. Step 1: Understand the Goal
  We need to find the square root of $1225$ . This means finding a number that, when multiplied by itself, equals $1225$ . We can write this as $\sqrt{1225}$ .
2. Step 2: Estimate the Range
  Numbers ending in 5 are good candidates. Let's check some: $30^2 = 900$ and $40^2 = 1600$ . Since $1225$ is between $900$ and $1600$ , its square root must be between $30$ and $40$ . Also, since $1225$ ends in $5$ , its square root must also end in $5$ .
3. Step 3: Factorize the Number
  To find the exact square root, let's break $1225$ into its prime factors. Dividing by $5$ , we get $1225 = 5 \times 245$ . Dividing $245$ by $5$ again, we get $245 = 5 \times 49$ . Since $49 = 7 \times 7$ , we have $1225 = 5 \times 5 \times 7 \times 7 = 5^2 \times 7^2$ .
4. Step 4: Calculate the Root
  Now that we have the prime factorization $1225 = 5^2 \times 7^2$ , we can easily find the square root. $\sqrt{1225} = \sqrt{5^2 \times 7^2} = \sqrt{5^2} \times \sqrt{7^2} = 5 \times 7 = 35$ .
5. Step 5: The answer
  So the answer is (B) 35. We found that $35 \times 35 = 1225$ , which means $35$ is the square root of $1225$ .
02
Evaluate $\sqrt[3]{729}$ .
5 steps · reveal one at a time
1. Step 1: Understand the Question
  We need to find the cube root of 729. This means finding a number that, when multiplied by itself three times, equals 729. We can write this as $\sqrt[3]{729}$ .
2. Step 2: Prime Factorization
  Let's find the prime factors of 729. We can see it's divisible by 3: $729 = 3 \times 243 = 3 \times 3 \times 81 = 3 \times 3 \times 3 \times 27 = 3 \times 3 \times 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$ . So, $729 = 3^6$ .
3. Step 3: Simplify the Expression
  Now we need to evaluate $\sqrt[3]{3^6}$ . We can rewrite $3^6$ as $(3^2)^3$ because $(a^m)^n = a^{mn}$ . So, $3^6 = (3^2)^3$ . This means we are looking for $\sqrt[3]{(3^2)^3}$ .
4. Step 4: Calculate the Value
  The cube root of a number cubed is the number itself. So, $\sqrt[3]{(3^2)^3} = 3^2$ . Calculating $3^2$ gives us $3 \times 3 = 9$ .
5. Step 5: The answer
  So the answer is (C) 9, because $9 \times 9 \times 9 = 81 \times 9 = 729$ .
03
$\sqrt{0.4}$ is closest to which value?
5 steps · reveal one at a time
1. Step 1: Convert to fraction
  It's often easier to work with square roots of decimals by first converting them to fractions. Let's rewrite $0.4$ as a fraction: $0.4 = \frac{4}{10}$ .
2. Step 2: Simplify the expression
  Now we have $\sqrt{\frac{4}{10}}$ . We can split this into $\frac{\sqrt{4}}{\sqrt{10}}$ . Since $\sqrt{4} = 2$ , this simplifies to $\frac{2}{\sqrt{10}}$ .
3. Step 3: Estimate $\sqrt{10}$
  To find the value, we need to estimate $\sqrt{10}$ . We know that $3^2 = 9$ and $4^2 = 16$ . So, $\sqrt{10}$ is slightly more than $3$ . A good approximation is $\sqrt{10} \approx 3.16$ .
4. Step 4: Calculate the value
  Now substitute the estimated value back into our expression: $\frac{2}{\sqrt{10}} \approx \frac{2}{3.16}$ . Performing this division gives us approximately $0.63$ .
5. Step 5: The answer
  So the answer is (B) 0.63, because our calculation $\sqrt{0.4} \approx 0.63$ matches this option exactly.
04
If $\sqrt{x} + \sqrt{81} = \sqrt{324}$ , find $x$ .
5 steps · reveal one at a time
1. Step 1: Simplify square roots
  First, simplify the known square roots in the equation. We know that $\sqrt{81} = 9$ and $\sqrt{324} = 18$ . This makes the equation much simpler to work with.
2. Step 2: Rewrite the equation
  Now, substitute these simplified values back into the original equation. The equation $\sqrt{x} + \sqrt{81} = \sqrt{324}$ becomes $\sqrt{x} + 9 = 18$ . This helps us isolate the term with $x$ .
3. Step 3: Isolate $\sqrt{x}$
  To find the value of $\sqrt{x}$ , subtract 9 from both sides of the equation. This gives us $\sqrt{x} = 18 - 9$ , which simplifies to $\sqrt{x} = 9$ . We are almost there!
4. Step 4: Solve for $x$
  Since we have $\sqrt{x} = 9$ , to find $x$ , we need to square both sides of the equation. So, $x = 9^2$ . Calculating this, we get $x = 81$ .
5. Step 5: The answer
  So the answer is (D) 81. The final value of $x$ is 81, which matches this option perfectly after simplifying the square roots and solving the equation.
05
Simplify $\dfrac{\sqrt{72} + \sqrt{800}}{\sqrt{18}}$ .
5 steps · reveal one at a time
1. Step 1: Simplify first term
  Let's start by simplifying the first square root in the numerator. We can write $\sqrt{72}$ as $\sqrt{36 \times 2}$ . Since $\sqrt{36} = 6$ , this simplifies to $6\sqrt{2}$ .
2. Step 2: Simplify second term
  Next, let's simplify the second square root in the numerator. We can write $\sqrt{800}$ as $\sqrt{400 \times 2}$ . Since $\sqrt{400} = 20$ , this simplifies to $20\sqrt{2}$ .
3. Step 3: Simplify denominator
  Now, let's simplify the square root in the denominator. We can write $\sqrt{18}$ as $\sqrt{9 \times 2}$ . Since $\sqrt{9} = 3$ , this simplifies to $3\sqrt{2}$ .
4. Step 4: Substitute and combine
  Substitute these simplified terms back into the original expression: $\dfrac{6\sqrt{2} + 20\sqrt{2}}{3\sqrt{2}}$ . Combine the terms in the numerator to get $\dfrac{26\sqrt{2}}{3\sqrt{2}}$ .
5. Step 5: The answer
  So the answer is (C) $\dfrac{26}{3}$ . We can cancel out the common factor of $\sqrt{2}$ from the numerator and the denominator, leaving us with $\dfrac{26}{3}$ .

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5 solved Square Roots and Cube Roots questions

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