Video: Simplifying Numerical Expressions Involving Cube Roots

Express ∛(2) − ∛(9) × ∛(−6) + 2∛(1/4) in its simplest form.

04:45

Video Transcript

Express the cube root of two minus the cube root of nine multiplied by the cube root of negative six plus two multiplied by the cube root of one-quarter in its simplest form.

Okay in order to actually express our expression in its simplest form, I’m actually gonna break it down into bits. The first section I’m looking at is the cube root of nine multiplied by the cube root of negative six. And I’m looking at this section because actually the two terms are multiplied together. So first of all, I actually want to begin by rewriting my terms.

So if I start with the cube root of nine, I can think of this as the cube root of three multiplied by three. And then if we have the cube root of negative six, we can think of this as the cube root of three multiplied by negative two. And then once again, we can actually rewrite because we’ve got the cube root of three multiplied by three and then we’ve also got another three in our another cube root, which is gonna give us the cube root of three multiplied by three multiplied by three multiplied by the cube root of negative two.

Well then, we actually know that the cube root of three multiplied by three multiplied by three is equal to the cube root of 27. So that’s just gonna be equal to three. But we also know, we would’ve got the three anyway because if you have the cube root of a number multiplied by itself three times, that is just going to be the number itself. Okay, so we’ve now got three multiplied by the cube root of negative two.

Then there’s another step that we can actually complete. And that’s because the cube root of negative two is gonna to be equal to negative the cube root of two. Because actually if you’ve got a negative inside the cube root, that means that our answer of the cube root of two is gonna be the same but just negative.

So therefore, we can say it’s equal to negative three the cube root of two. Okay, great! So also at this point, we can double check that actually yes we’ve got the cube root of two. And that works because if we look at the first term, that’s also the cube of two. I’m gonna need actually a similar cubed root because we’re actually looking to simplify this expression.

Okay, so we can now move on to the final term and work with this. We’ve got two multiplied by the cube root of a quarter. Well then, we can use an exponent rule, which is that the 𝑎th root of 𝑥 is equal to 𝑥 to the power of one over 𝑎. So therefore we get equal to two multiplied by a quarter to the power of third. And then we’ll get two multiplied by and, then now because a quarter is the same as a half squared, we’re gonna have a half squared to the power of a third.

And then we apply another exponent rule, which is actually that 𝑥 to the power of 𝑎 in parentheses to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. So this gives us that two multiplied by half to the power of two over three. And then we can actually apply another exponent rule, which tells us that one over 𝑥 to the power of 𝑎 is equal to 𝑥 to the power of negative 𝑎. So now we actually get this power expression equal to two multiplied by two to the power of negative two over three.

And then we can actually add our exponents because we’re actually multiplying two terms. So we’ll have one add negative two over three. So we get this is equal to two to the power of a third. And then taking a look back at our first exponent rule, we can say that this is equal to the cubed root of two. Okay, great! So we’ve now dealt with individual parts. Let’s put it all back together so that we can fully simplify our expression.

So we have the cube root of two. And then we get plus three cube root two. And we get this because in the original expression we had minus cube root nine. And then we know that cube root nine is equal to negative three cube root two. So minus a negative gives us a positive. And then finally, we have plus cube root two. And that’s because that’s what our two cube root of a quarter became when we simplified that.

So therefore, we can say that cube root of two minus cube root of nine multiplied by the cube root of negative six plus two cube root of a quarter in its simplest form is equal to five the cube root of two. And we got that because cube root of two plus three cube root of two plus another cube root of two will give us five cube root of two.

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