The cube root of a number 𝑧 is four 𝑥 cubed 𝑦 to the power of 15. Write this root in the form the cube root of 𝑧.
Well, if we look at the question and the information that we’ve got, we know that the cube root of 𝑧 is equal to four 𝑥 cubed 𝑦 to the power of 15. However, what we wanna find is actually the 𝑧-value that goes within our cube root. So therefore, in order to actually find out what the value of 𝑧 is, we’re gonna need to do the inverse of our cube root, which is actually cube.
So therefore, what we’re gonna do is cube both sides of our equation. So we’re gonna get 𝑧 is equal to four 𝑥 cubed 𝑦 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 𝑥 to the power of 𝑎 to the power of 𝑏, this is equal to 𝑥 to the power of 𝑎𝑏. So we actually multiply the powers.
What I’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, we’re gonna have four cubed. And then we’re gonna have this multiplied by 𝑥 cubed cubed and then multiplied by 𝑦 to the power of 15 cubed. And I’ve done this, like I said, I wouldn’t usually put this line in but just to show exactly how we reach each part of our term.
So therefore, we’re gonna have 𝑧 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 𝑥 to the power of nine. And that’s because we had 𝑥 power three to the power of three.
Well, if you have 𝑥 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 that’s where we got our 𝑥 power of nine. And then, finally, this is multiplied by 𝑦 to the power of 45. We got this because we had 𝑦 to the power of 15 to power three.
Again multiply our exponents. Three by 15 gives us 45. So we’ve now got that 𝑧 is equal to 64𝑥 to power nine 𝑦 to power of 45. But have we finished here? We found 𝑧. 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 𝑧. So therefore, we can say that four 𝑥 cubed 𝑦 to the power of 15 is equal to the cube root of 64𝑥 to the power of nine 𝑦 to the power of 45.