### Video Transcript

The cube root of a number ๐ง is four ๐ฅ cubed ๐ฆ to the power of 15. Write this root in the form the cube root of ๐ง.

Well, if we look at the question and the information that weโve got, we know that the cube root of ๐ง is equal to four ๐ฅ cubed ๐ฆ to the power of 15. However, what we wanna find is actually the ๐ง-value that goes within our cube root. So therefore, in order to actually find out what the value of ๐ง is, weโre gonna need to do the inverse of our cube root, which is actually cube.

So therefore, what weโre gonna do is cube both sides of our equation. So weโre gonna get ๐ง is equal to four ๐ฅ cubed ๐ฆ to the power of 15 all cubed. Well, here we can actually use one of our exponent rules. And that rule is that if we have ๐ฅ to the power of ๐ to the power of ๐, this is equal to ๐ฅ to the power of ๐๐. So we actually multiply the powers.

What Iโm gonna do is actually split it into parts to actually help us see how we reduce that on our particular right-hand side of the equation. So first, weโre gonna have four cubed. And then weโre gonna have this multiplied by ๐ฅ cubed cubed and then multiplied by ๐ฆ to the power of 15 cubed. And Iโve done this, like I said, I wouldnโt usually put this line in but just to show exactly how we reach each part of our term.

So therefore, weโre gonna have ๐ง is equal to 64. And this is because four cubed is four multiplied by four multiplied by four, which is 64. And then we have this multiplied by ๐ฅ to the power of nine. And thatโs because we had ๐ฅ power three to the power of three.

Well, if you have ๐ฅ power three to the power of three, we use the exponent rule. So we multiply the exponent. So three multiplied by three gives us nine. So thatโs where we got our ๐ฅ power of nine. And then, finally, this is multiplied by ๐ฆ to the power of 45. We got this because we had ๐ฆ to the power of 15 to power three.

Again multiply our exponents. Three by 15 gives us 45. So weโve now got that ๐ง is equal to 64๐ฅ to power nine ๐ฆ to power of 45. But have we finished here? We found ๐ง. Is that what we want? Well, no, if we take a look back at the question, we can say that we want the answer in the form the cube root of ๐ง. So therefore, we can say that four ๐ฅ cubed ๐ฆ to the power of 15 is equal to the cube root of 64๐ฅ to the power of nine ๐ฆ to the power of 45.