Tag: estimation of cube roots

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

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

  1. $2a$
  2. $2b$
  3. $3a$
  4. $4a$
Reveal answer Fill a bubble to check yourself
A Correct answer
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, $x^3+3bx=2a$.
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

      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 $\sqrt[3]{1333}$ $= 10.99$