Name: Finding Cube of a Rational Number
Uploaded: 2019-07-19
Duration: 1 min 43 s
Description: We have learned about the cube root of a negative number. In this video, we will learn how to find the cube root of a rational number. Explore the video to l...

Question

What is the smallest number by which

8192

must be divided so that quotient is a perfect cube? Also, find the cube root of the quotient so obtained.

Accepted Answer

By using prime factorization method, resolving the given number into prime factors, we get,

$8192 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2$

Since, one triple remain incomplete, hence $8192$ is not a perfect cube.

So, we divide $8192$ by $2$ to make its quotient a perfect cube.

$8192 \div 2 = 4096$

$4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{3} \times 2^{3} \times 2^{3} \times 2^{3}$

Cube root of $4096 = \sqrt[3]{4096} = \sqrt[3]{(2^{3} \times 2^{3} \times 2^{3} \times 2^{3})} = 2 \times 2 \times 2 \times 2 = 16$

$∴$ $2$ is the smallest number by which $8192$ must be divided so that quotient is a perfect cube and the cube root of $4096$ is $16$ .

What is the smallest number by which $8192$ must be divided so that quotient is a perfect cube? Also, find the cube root of the quotient so obtained.

Important Questions on Cubes and Cube Roots

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