Question

Find the smallest number by which 256 must be multiplied to obtain a perfect cube.​

Answer 1

Answer:

2

Step-by-step explanation:

In order to find the smallest number by which 256 must be multiplied to obtain a perfect cube, firstly we need to resolve the given number into prime factors.

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By prime factorization, we get :

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$\large\begin{array}{c | c} \underline{\sf{2}} & \underline{ \sf{256}} \\\underline{\sf{2}} & \underline{ \sf{128}} \\ \underline{\sf{2}} & \underline{ \sf{ \: 64 \: }} \\\underline{\sf{ 2 }} & \underline{ \sf{ \: 32 \: }} \\ \underline{\sf{2}} & \underline{ \sf{ \: 16 \: }} \\ \underline{\sf{2}} & \underline{ \sf{ \: \: 8 \: \: }} \\ \underline{\sf{2}} & \underline{ \sf{ \: \: 4 \: \: }} \\ \underline{\sf{2}} & \underline{ \sf{ \: \: 2 \: \: }} \\ &1 \end{array}$

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$\longmapsto$ 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

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Grouping into triplets of common factors.

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$\longmapsto \sf {\sqrt[3]{256} }$ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

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We get that we need 2 to make one more triplet of 2. So, 2 is the smallest number by which 256 must be multiplied to obtain a perfect cube.

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Therefore, 2 is the required answer.

More about cubes :

• Cubes of all odd numbers are odd.

• Cubes of all even numbers are even.

• Cube of a negative integer is negative.

How to find cube root of any number by prime factorization ?

• Resolve the given number into prime factors.

• Group together triplets of the same prime factor.

• Pick out each factor from the triplets of the same prime factor.

• The product of those picked factors will be the cube root of the given number.