Tag: estimation of cube roots

Questions Related to estimation of cube roots

If $x = \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$ then $x^{3} + 3bx = $ ____________.

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

  1. $2a$

  2. $2b$

  3. $3a$

  4. $4a$

Show answer
Correct Option: A
Explanation:
Given,
$x = \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$.......(1).
Now cubing both sides we get,
$x^3=a+\sqrt{a^2-b^3}+a-\sqrt{a^2-b^3}-3$$\sqrt [3]{a + \sqrt {a^{2} - b^{3}}}  \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$$( \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}})$
or, $x^3=2a-3bx$ [ Using (1)and $ (\sqrt [3]{a + \sqrt {a^{2} - b^{3}}})(\sqrt [3]{a - \sqrt {a^{2} - b^{3}}})=\sqrt[3]{a^2-(a^2-b^3)}=b$]
or, $x^3+3bx=2a$.

Estimate the cube root of the number $23.$

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

  1. $2.6$

  2. $2.1$

  3. $2.8$

  4. $1.4$

Show answer
Correct Option: C
Explanation:
$\sqrt[3]{\dfrac{23 \times 1000}{1000}}$
$\sqrt[3]{23000} \div 10$
Now, find the closest cube root of $23000.$
$28^3 = 21952,$ therefore we can say that $\sqrt[3]{23}$ $\sim$ $\dfrac{28}{10} = 2.8$

Find the cube root of the number $120.$

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

  1. $4.1$

  2. $4.2$

  3. $4.7$

  4. $4.9$

Show answer
Correct Option: D
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 $'D'.$

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

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

      1. $2.08$

      2. $2.19$

      3. $2.34$

      4. $2.51$

      Show answer
      Correct Option: A
      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 $\sqrt[3]{9}$ $\sim$ $\dfrac{208}{100} \sim 2.08$

      State whether true or false:
      If $x^3 = 11$, then $x=\sqrt[3] {11}$
      maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

      1. True

      2. False

      Show answer
      Correct Option: A
      Explanation:
      TRUE:
      Odd exponents preserve the sign of the original expression. Therefore, if ${x}^{3}$ is positive, then $x$ must itself be positive. IF ${x}^{3}=11$, then $x$ must be $\sqrt [ 3 ]{ 11 } $

      What is the approximate value of the cube root of the number $12?$

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

      1. $2.6$

      2. $2.1$

      3. $2.8$

      4. $2.2$

      Show answer
      Correct Option: D
      Explanation:

      $\sqrt[3] {\dfrac{12\times 1000}{1000}}$
      $\sqrt[3]{12000} \div 10$
      Now find the closest cube root of $12000.$
      $22^3 = 10648,$ therefore we can say that $\sqrt[3]{12}$ $\sim$ $\dfrac{22}{10} = 2.2$

      If $\displaystyle \sqrt[3]{3\left ( \sqrt[3]{x}-\frac{1}{\sqrt[3]{x}} \right )}=2$ then $\displaystyle \sqrt[3]{x}+\frac{1}{\sqrt[3]{x}}=$_________

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

      1. $\displaystyle \frac{8}{3}$

      2. 0

      3. 1

      4. -1

      Show answer
      Correct Option: A
      Explanation:

      $\displaystyle \sqrt[3]{3\left ( \sqrt[3]{x}-\frac{1}{\sqrt[3]{x}} \right )}=2$ 
      Let us assume $\left( \sqrt [ 3 ]{ x } -\frac { 1 }{ \sqrt [ 3 ]{ x }  }  \right) $=a
      So, $\sqrt [ 3 ]{ 3a } =2$
      Taking  cube both sides,
      $3a=8$
      $a=\frac { 8 }{ 3 } $
      So, $\left( \sqrt [ 3 ]{ x } -\frac { 1 }{ \sqrt [ 3 ]{ x }  }  \right) =\frac { 8 }{ 3 } $

      The simplest form of $\sqrt[3]{768}$ is

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

      1. $2\sqrt[3]{12}$

      2. $4\sqrt[3]{12}$

      3. $3\sqrt{12}$

      4. $3\sqrt[3]{6}$

      Show answer
      Correct Option: B
      Explanation:

      $\sqrt[3]{768}=\sqrt[3]{2^3\times2^3\times2\times2\times3}=2\times2\sqrt[3]{12}=4\sqrt[3]{12}$

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

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

      1. $1$

      2. $2$

      3. $3$

      4. $4$

      Show answer
      Correct Option: D
      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 $4$.

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

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

      1. $10.99$

      2. $20.10$

      3. $12.45$

      4. $10.56$

      Show answer
      Correct Option: A
      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 $\sqrt[3]{1333}$ $= 10.99$