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

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

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

  2. 0

  3. 1

  4. -1
Reveal answer Fill a bubble to check yourself
A Correct answer
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 } $

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 $9$.