Question Video: Evaluating Algebraic Expressions Involving Roots and Exponents Mathematics

Given that 𝑥 = ²√6 − 10 and 𝑦 = ∛6 + 10, find the value of 𝑥² + 𝑦².

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Video Transcript

Given that 𝑥 equals cube root of six minus 10 and 𝑦 equals cube root of six plus 10, find the value of 𝑥 squared plus 𝑦 squared.

So the first stage of this problem is to substitute in our value for 𝑥 and 𝑦 into the expression 𝑥 squared plus 𝑦 squared. And when we do that, we get cube root of six minus 10 all squared plus cube root of six plus 10 all squared. And that’s because cube root of six minus 10 is our 𝑥 and cube root of six plus 10 is our 𝑦.

So now, the next step is to actually expand the parenthesis. So I’ve rewritten it as cube root of six minus 10 multiplied by cube root of six minus 10 plus cube root of six plus 10 multiplied by cube root of six plus 10.

So to enable us to expand the parenthesis, what we need to do is remind ourselves of this rule. And that’s if you have the cube root of 𝑎 multiplied by the cube root of 𝑏, it’s equal to the cube root of 𝑎 multiplied by 𝑏 or 𝑎𝑏.

So therefore, if we multiply cube root of six by cube of six, we’re gonna get cube root of 36. And that’s because six multiplied by six is 36. Then, we’ll have minus 10 cube root of six cause that’s cube root of six multiplied by negative 10. And then, we’ll get another minus 10 cube root of six then finally plus 100 because negative 10 multiplied by negative 10 gives us positive 100. Then, we’re gonna have plus another cube root of 36 cause again we have cube root of six multiplied by cube root of six then plus 10 cube root of six then plus another 10 cube root of six and then plus 100 because this time have positive 10 multiplied by positive 10 which gives us positive 100.

So now, what we want to do is actually collect like terms and simplify. Well, first of all, we’ve got cube root of 36 plus cube root of 36. So therefore, this’s gonna give us two cube root of 36. Then, we’ve got negative 10 cube root of six plus 10 cube root of six, so these cancel out, and then another negative 10 cube root of six plus 10 cube root of six. And if we have negative cube root of six and add 10 to it, again, we get zero. So we can cancel these ones out. And then, finally, we’ve got plus 200. And that’s because we had positive 100 add 100, which is 200.

So therefore, we can say that given that 𝑥 is equal to cube root of six minus 10 and 𝑦 is equal to the cube root of six plus 10, the value of 𝑥 squared plus 𝑦 squared is two cube root of 36 plus 200.

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