Mathematics · Quantitative Aptitude

Surds and Indices

408 Questions

Surds and indices questions focus on evaluating square roots, cube roots, and fractional exponents. These topics are a core part of the quantitative aptitude section in many competitive exams. Practicing these problems builds speed and accuracy for solving numerical equations.

Square roots evaluationCube roots calculationExponents and powersFractional exponentsSurds multiplication

Surds and Indices Questions

Multiple choice

The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x2 – N = 0. If i denotes the iteration index, the correct iterative scheme will be
  1. $X_{i+1} = \frac{1}{2}\Big(X_i + \frac{N}{X_i}\Big)$
  2. $X_{i+1} = \frac{1}{2}\Big(X_i^2 + \frac{N}{X_i^2}\Big)$
  3. $X_{i+1} = \frac{1}{2}\Big(X_i + \frac{N^2}{X_i}\Big)$
  4. $X_{i+1} = \frac{1}{2}\Big(X_i - \frac{N}{X_i}\Big)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Newton-Raphson iteration for f(x) = 0 is x_{i+1} = x_i - f(x_i)/f'(x_i). For f(x) = x^2 - N, f'(x) = 2x. Thus, x_{i+1} = x_i - (x_i^2 - N)/(2x_i) = (2x_i^2 - x_i^2 + N)/(2x_i) = (x_i^2 + N)/(2x_i) = 1/2 * (x_i + N/x_i).

Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

Find the cube root of the number $120.$

  1. $4.1$
  2. $4.2$
  3. $4.7$
  4. $4.9$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
We use the Babylonian Algorithm for cube roots here
According to the algorithm, the cube root is given by the formula 
$x _{n+1}=\dfrac{\left (2x _n+\left (\dfrac N{x _{n^2}}\right )\right )}{3}$
where,
  • $N$ is the number for which cube root is to be found
  • $x _{n}$ is the initial approximation of the cube root
  • $x _{n+1}$ is the subsequent improvement on the cube root 

In this case,
$N = 120$
    $x _0 =4$ since $4^3<40 <5^3$

      $ \therefore$ $x _1 = \dfrac{\left ((2\times4)+\left (\dfrac {120} {4^2}\right )\right )}{3} = \dfrac{\left (8+\left (\dfrac {120}{16}\right )\right )}{3}=4.9$

      $\Rightarrow x _2 =\dfrac{ \left (2\times4.9+\left (\dfrac {120}{(4.9)^2}\right )\right )}{3} = \dfrac{\left (9.8+\left (\dfrac {120}{24.01} \right )\right )}{3}= \dfrac{(9.8+4.99)}{3} = 4.9$

      We can see the value stabilizes around $4.9$. 

      Hence the answer is $'D'.$
      Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      What is the approximate value of the cube root of the number $9?$

      1. $2.08$
      2. $2.19$
      3. $2.34$
      4. $2.51$
      Reveal answer Fill a bubble to check yourself
      A Correct answer
      Explanation

      First multiply and divide by $1,000,000,$ we get
      $\sqrt[3]{\dfrac{9\times 1000,000}{1000,000}}$
      $\sqrt[3]{9,000,000} \div 100$
      Now find the closest cube root of $9,000,000.$
      $208^3 = 8,998,912,$ therefore we can say that $\sqrt[3]{9}$ $\sim$ $\dfrac{208}{100} \sim 2.08$

      Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      Find the value of cube root of the number $45$. (Round off your number to the nearest whole number)

      1. $1$
      2. $2$
      3. $3$
      4. $4$
      Reveal answer Fill a bubble to check yourself
      D Correct answer
      Explanation

      We need to find value of $\sqrt[3]{45}$
      Take, $n = 45$, choose any starting value of $x$.
      So, $3^3$ is $27 < 45$
      So, $x = 3$
      $x _\text{next} =$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
      $x _\text{next} = $ $\dfrac{2}{3}3+\dfrac{45}{3\times 3^2}$
      $x _\text{next} = 3.6666$
      So, the nearest whole number for the cube root $45$ is $4$.

      Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      Estimate the value of cube root of the number $1333$.

      1. $10.99$
      2. $20.10$
      3. $12.45$
      4. $10.56$
      Reveal answer Fill a bubble to check yourself
      A Correct answer
      Explanation

      We need to find $\sqrt[3]{1333}$
      Take, $n = 1333$, choose any starting value of $x$.
      So, $11^3$ is $1331 < 1333$
      So, $x = 11$
      $x _\text{next}$ $=$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
      $x _\text{next}$ $=$ $\dfrac{2}{3}11+\dfrac{1331}{3\times 11^2}$
      $x _\text{next}$ $= 10.999 $    ....(1)
      Assume $x = 10.99$
      $x _\text{next} =$ $\dfrac{2}{3}10.99+\dfrac{1331}{3\times 10.99^2}$
      $x _\text{next} = 10.99$     ....(2)
      Since we are getting $10.99$ in (1) and (2)
      So, the approximate value of $\sqrt[3]{1333}$ $= 10.99$

      Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      Find the cube root of the number 514.

      1. 8.0104

      2. 8.1104

      3. 8.2104

      4. 8.3104
      Reveal answer Fill a bubble to check yourself
      A Correct answer
      Explanation
      We use the Babylonian Algorithm for cube roots here
      According to the algorithm, the cube root is given by the formula 
      $x _{n+1}=\dfrac{(2x _n+(N/x _{n^2}))}{3}$
      where
      • $N$ is the number for which cube root is to be found
      • $x _{n}$ is the initial approximation of the cube root
      • $x _{n+1}$ is the subsequent improvement on the cube root
      In this case,
      $N = 514$
        $x _0 =8$ since, $8^3<514 <9^3$

          $ \therefore$ $x _1 = \dfrac{((2\times8)+(514 /8^2))}{3} = \dfrac{(16+(514/64))}{3}=8.0104$

          $\Rightarrow x _2 = \dfrac{((2\times8.0104+(514/(8.0104)^2))}{3} = \dfrac{(16.02.08+((514/64.1666))}{3}=8.0104$

          We can see the value stabilizes around $8.0104$. Hence the answer is A.

          Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

          Estimate the cube root of the number $40.$

          1. $3.1$
          2. $3.2$
          3. $3.4$
          4. $3.9$
          Reveal answer Fill a bubble to check yourself
          C Correct answer
          Explanation

          We use the Babylonian Algorithm for cube roots here

          According to the algorithm, the cube root is given by the formula 
            $x _{n+1}=\dfrac{(2x _n+(N/x _{n^2}))}{3}$
            where,
            • $N$ is the number for which cube root is to be found
            • $x _{n}$ is the initial approximation of the cube root
            • $x _{n+1}$ is the subsequent improvement on the cube root
            • In this case 
            $N = 40$
              $x _0 =3$ since $3^3<40 <4^3$


                $ \therefore$ $x _1 = \dfrac{((2\times3)+(40 /3^2))}{3} = \dfrac{(6+(40/9))}{3}=3.4$

                $\Rightarrow x _2 = \dfrac{(2\times3.4+40/(3.4)^2)}{3} = \dfrac{(6.8+(40/11.56))}{3}= \dfrac{(6.8+3.46)}{3} = 3.4$

                We can see the value stabilizes around 3.4. Hence, the answer is C.

              Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

              Find the value of cube root of the number $1290$. (Round off your number to the nearest whole number)

              1. $9.2$
              2. $10.2$
              3. $10.96$
              4. $11$
              Reveal answer Fill a bubble to check yourself
              D Correct answer
              Explanation

              We need to find value of $\sqrt[3]{1290}$
              Take, $n = 1290$, choose any starting value of $x$.
              So, $10^3$ is $1000 < 1290$
              So, $x = 10$
              $x _\text{next} =$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
              $x _\text{next} =$ $\dfrac{2}{3}10+\dfrac{1290}{3\times 10^2}$
              $x _\text{next} =  10.966$
              So, the nearest whole number for the cube root $1290$ is $11$.

              Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

              Find the value of cube root of the number $6860$. (Round off your number to the nearest hundredth)

              1. $19.003$
              2. $19.00$
              3. $19.32$
              4. $19.008$
              Reveal answer Fill a bubble to check yourself
              B Correct answer
              Explanation

              We need to find value of $\sqrt[3]{6860}$
              Take, $n = 6860$, choose any starting value of $x$.
              So, $19^3$ is $6859 < 6860$
              So, $x = 19$
              $x _\text{next} =$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
              $x _\text{next} = $ $\dfrac{2}{3}19+\dfrac{6860}{3\times 19^2}$
              $x _\text{next} = 19.00085 $    ....(1)
              So, the approximate value of $\sqrt[3]{6860}$ nearest hundredth place is $19.00$.

              Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

              Find the value of cube root of the number $823$. (Round off your number to the nearest whole number)

              1. $8$
              2. $7$
              3. $9$
              4. $10$
              Reveal answer Fill a bubble to check yourself
              C Correct answer
              Explanation

              We need to find value of $\sqrt[3]{823}$
              Take, $n = 823$, choose any starting value of $x$.
              So, $9^3$ is $729 < 823$
              So, $x = 9$
              $x _\text{next} =$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
              $x _\text{next} =$ $\dfrac{2}{3}9+\dfrac{823}{3\times 9^2}$
              $x _\text{next} = 9.386$
              So, the nearest whole number for the cube root $823$ is $9$.