Video: Finding an Algebraic Expression given Its Cubic Root

The cube root of a number ๐‘ง is 4๐‘ฅยณ๐‘ฆยนโต. Write this root in the form โˆ›๐‘ง.

02:29

Video Transcript

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

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

So therefore, what weโ€™re gonna do is cube both sides of our equation. So weโ€™re gonna get ๐‘ง is equal to four ๐‘ฅ cubed ๐‘ฆ 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 ๐‘ฅ to the power of ๐‘Ž to the power of ๐‘, this is equal to ๐‘ฅ to the power of ๐‘Ž๐‘. So we actually multiply the powers.

What Iโ€™m 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, weโ€™re gonna have four cubed. And then weโ€™re gonna have this multiplied by ๐‘ฅ cubed cubed and then multiplied by ๐‘ฆ to the power of 15 cubed. And Iโ€™ve done this, like I said, I wouldnโ€™t usually put this line in but just to show exactly how we reach each part of our term.

So therefore, weโ€™re gonna have ๐‘ง 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 ๐‘ฅ to the power of nine. And thatโ€™s because we had ๐‘ฅ power three to the power of three.

Well, if you have ๐‘ฅ 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 thatโ€™s where we got our ๐‘ฅ power of nine. And then, finally, this is multiplied by ๐‘ฆ to the power of 45. We got this because we had ๐‘ฆ to the power of 15 to power three.

Again multiply our exponents. Three by 15 gives us 45. So weโ€™ve now got that ๐‘ง is equal to 64๐‘ฅ to power nine ๐‘ฆ to power of 45. But have we finished here? We found ๐‘ง. 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 ๐‘ง. So therefore, we can say that four ๐‘ฅ cubed ๐‘ฆ to the power of 15 is equal to the cube root of 64๐‘ฅ to the power of nine ๐‘ฆ to the power of 45.

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