# Question Video: Simplifying Algebraic Fractions Using Properties of Roots and Exponents Mathematics

Simplify ∛(∛512/𝑥(63)).

03:44

### Video Transcript

Simplify the cube root of the cube root of 512 over 𝑥 to the power of 63.

Here we have a rather interesting problem which involves two different cube roots. The way that we solve this problem is by simplifying the innermost cube root first. So we simplify the cube root of 512 over 𝑥 to the power 63. To do this, we’ll need to apply some exponent rules. The first rule we can apply is that for any real values of the 𝑛th root of 𝑎 and the 𝑛th root of 𝑏, then the 𝑛th root of 𝑎 over the 𝑛th root of 𝑏 is equivalent to the 𝑛th root of 𝑎 over 𝑏.

And so we know that the cube root of 512 over 𝑥 to the power 63 is equal to the cube root of 512 over the cube root of 𝑥 to the 63rd power. We can then simplify the numerator and denominator in turn. So let’s work out the cube root of 512. Remember that to work out the cube root of 512, that means that we’re looking for the value which when written three times and multiplied gives us an answer of 512. That would be eight. And so the numerator simplifies to eight.

Next, let’s see how we simplify the cube root of 𝑥 to the 63rd power. To do this, we can use another exponent rule. That is that the 𝑛th root of 𝑎 is equal to 𝑎 to the power of one over 𝑛. The cube root here can therefore be written as the power of one-third. But we can simplify this denominator 𝑥 to the power of 63 to the power of one-third even further. We remember that 𝑎 to the power 𝑥 to the power 𝑦 is equal to 𝑎 to the power 𝑥𝑦. Therefore, on the denominator, we multiply the exponents of 63 and one-third. 63 times one-third is 21, leaving us with 𝑥 to the power 21 on the denominator.

Remember, at this point, we haven’t fully simplified the question. We have only simplified the cube root of 512 over 𝑥 to the power 63 as eight over 𝑥 to the 21st power. We still need to take a further cube root of this expression. In order to simplify this expression, we can apply the first exponent rule again. Therefore, on the right-hand side, we have the cube root of eight over the cube root of 𝑥 to the power 21. Taking the numerator and denominator in turn, we work out that the cube root of eight is two.

Then to simplify the cube root of 𝑥 to the power of 21, we apply the second exponent rule. We can then write this as 𝑥 to the power 21 to the power one-third. Applying the third exponent rule, we know that we can multiply the indices of 21 and one-third. And 21 times one-third is seven, which leaves 𝑥 to the power seven on the denominator. We cannot simplify this any further. So we can give the answer that the cube root of the cube root of 512 over 𝑥 to the 63rd power is two over 𝑥 to the seventh power.