Question

Is 40500 a perfect cube? If not (a) by what number should 40500 be multiplied to obtain a perfect cube?​

Answer 1

SOLUTION-;

We need to determine what number should be multiplied to 40500 to obtain a perfect cube.

Let’s first do prime factorization of 40500

40500 = 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5 x 5

$= {2}^{2} \times {3}^{2} \times 3 \times {5}^{3}$

For perfect cube we need all the powers must be 3.

So if we multiply one 2 and two threes on both side we get

$40500 \times 2 \times 3 \times 3 = {2}^{3} \times {3}^{3} \times {3}^{3} \times {5}^{3}$

So we get 40500 x 18 = 729000 and cube root of 729000 = 2 x 3 x 3 x 5 = 90.

Hence 18 should be multiplied to 40500 to get perfect cube that is 729000.

Answer 2

Given

• 40500

To Find

• 40500 is perfect cube or not
• If not by what number should 40500 be multiplied to obtain a perfect cube?

Solution

• By prime factorization

$\: \: \: \: \: \: \: \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}{\begin{array}{c | } \sf 2 & \sf 40500 \\ \hline \sf 2 & \sf 20250 \\ \hline\sf 5 & \sf 10125 \\ \hline\sf 5 & \sf 2025 \\ \hline\sf 5 & \sf 405\\ \hline\sf 3 & \sf 81 \\ \hline\sf 3 & \sf 27 \\ \hline\sf 3 & \sf 9 \\ \hline\sf 3& \sf 3 \\ \hline\sf & \sf 1 \\ \end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

${\sf \footnotesize 40500 = 2×2×5×5×5×3×3×3×3 }$

${\sf \footnotesize 40500 = 2² × 5³ × 3³ ×3 }$

• $\footnotesize \text{Multiply both sides by 2 and 3² for Making Power three On RHS. }$

${\sf \footnotesize 40500 × 2 × 3² = 2² × 5³ × 3³ ×3 × 2 × 3² }$

${\sf \footnotesize 40500 × 2 × 9 = 2³ × 5³ × 3³ }$

${\sf \footnotesize 40500 × 18 = 2³ × 5³ × 3³ }$

${\sf \footnotesize 729000 = 2³ × 5³ × 3³ }$

Cube root of 729000 = 30

Hence,

• We need to multiply 40500 by 18 to obtain a perfect cube that is 72900.