Question

Find the cube root of 17576 using factorization.

Answer 1

Hint: Divide the number 17576 into 2 groups. From the group find the unit’s place of the cube root. From the second group find the ten’s place of the cube root. Write the ten’s place and unit place together to get the cube root.

Complete step-by-step answer:
Given us the number 17576, now let us make the 17576 into 2 groups. The first group will be (576) and the \[{{2}^{nd}}\] group (17).

17 – Second group.

576 – First group.

Now let us find the unit digit of cube root.

The unit digit of cube root of 17576 = unit digit of cube root of 576.
576 ends with 6, so its cube root will end with 6.

For example, \[{{6}^{3}}=216,{{16}^{3}}=4096\], they all end with 6.
So, we found the unit digit of cube root as 6.

Now let us take the \[{{2}^{nd}}\] group which is 17.

We know that \[{{2}^{3}}\] is 8. Similarly, \[{{3}^{3}}\] is 27. Therefore, 17 lies between \[{{2}^{3}}\] and \[{{3}^{3}}\].
\[\begin{align}
  & \Rightarrow 8<17<27 \\
 & \Rightarrow {{2}^{3}}<17<{{3}^{3}} \\
\end{align}\]

Thus we can take the similar number.
So let us take 2.
Now let us put in the ten’s place of cube root.

Thus we got 6 in unit place and 2 in ten’s place.
\[\therefore \] Cube root of 17576 = 26

Note:
We can also find cube roots by prime factorization.
\[\begin{align}
  & 17576=2\times 8788 \\
 & =2\times 2\times 4394 \\
 & =2\times 2\times 2\times 2197 \\
 & =2\times 2\times 2\times 13\times 13\times 13 \\
 & =\left( 2\times 13 \right)\times \left( 2\times 13 \right)\times \left( 2\times 13 \right) \\
 & =26\times 26\times 26 \\
 & \therefore \sqrt[3]{17576}=\sqrt[3]{26}=26 \\
\end{align}\]