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Estimation of cube root - class-VIII

Description: estimation of cube root
Number of Questions: 16
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Tags: maths cube and cube root squares, square roots, cubes, cube roots cubes and cube roots
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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 $\sqrt[3]{1333}$ $= 10.99$

      Find the cube root of the number 514.

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

      1. 8.0104

      2. 8.1104

      3. 8.2104

      4. 8.3104

      Show answer
      Correct Option: A
      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.

          Estimate the cube root of the number $40.$

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

          1. $3.1$

          2. $3.2$

          3. $3.4$

          4. $3.9$

          Show answer
          Correct Option: C
          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.

              Find the value of cube root of the number $1290$. (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. $9.2$

              2. $10.2$

              3. $10.96$

              4. $11$

              Show answer
              Correct Option: D
              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 $11$.

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

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

              1. $19.003$

              2. $19.00$

              3. $19.32$

              4. $19.008$

              Show answer
              Correct Option: B
              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 $19.00$.

              Find the value of cube root of the number $823$. (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. $8$

              2. $7$

              3. $9$

              4. $10$

              Show answer
              Correct Option: C
              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 $9$.

              Find the value of cube root of the number $2486$. (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. $11$

              2. $12$

              3. $13$

              4. $14$

              Show answer
              Correct Option: D
              Explanation:

              We need to find value of $\sqrt[3]{2486}$
              Take, $n = 2486$, choose any starting value of $x$.
              So, $13^3$ is $2197 < 2486$
              So, $x = 13$
              $x _\text{next} =$ $\dfrac{2}{3}x+\dfrac{n}{3x^2}$
              $x _\text{next} =$ $\dfrac{2}{3}13+\dfrac{2486}{3\times 13^2}$
              $x _\text{next} = 13.56$
              So, the nearest whole number for the cube root $2486$ is $14$.

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