Question

Find the cube root of the following number −729 × −15625 .

Answer 1

Property:
For any two integers a and b,

\[\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}\]

From the above property, we have:​

\[\sqrt[3]{- 729 \times - 15625}\]

\[ = \sqrt[3]{- 729} \times \sqrt[3]{- 15625}\]

\[= - \sqrt[3]{729} \times - \sqrt[3]{15625}\]     (For any positive integer x,

\[\sqrt[3]{- x} = - \sqrt[3]{x}\]

Cube root using units digit:
Let us consider the number 15625.
The unit digit is 5; therefore, the unit digit in the cube root of 15625 will be 5.
After striking out the units, tens and hundreds digits of the given number, we are left with 15.
Now, 2 is the largest number whose cube is less than or equal to 15 (\[\left( 2^3 < 15 < 3^3 \right)\].

Therefore, the tens digit of the cube root of 15625 is 2.

∴ \[\sqrt[3]{15625} = 25\]

Also

\[\sqrt[3]{729} = 9, because 9 \times 9 \times 9 = 729\]

Thus

\[\sqrt[3]{- 729 \times - 15625} = - \sqrt[3]{729} \times - \sqrt[3]{15625} = - 9 \times - 25 = 225\]