Question Video: Evaluating Numerical Expressions Involving Cubes and Cube Roots Mathematics

If 𝑥 = ∛375 and 𝑦 = ∛81, find the value of (𝑥 + 𝑦)³.

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Video Transcript

If 𝑥 is equal to the cube root of 375 and 𝑦 is equal to the cube root of 81, find the value of 𝑥 plus 𝑦 all cubed.

In this question, we are given the values of 𝑥 and 𝑦 as radicals and are asked to evaluate an expression involving 𝑥 and 𝑦. We can begin by substituting the given values into the expression to obtain the cube root of 375 plus the cube root of 81 all cubed. At this point, there are two options for evaluating the expression. Either we can attempt to simplify the expression inside the parentheses or we can expand the cube of the terms in the parentheses.

Of course, expanding the cube will make the expression quite complicated. So, we will start by trying to simplify the expression inside the parentheses. To do this, we need to simplify each cube root. We can do this by recalling that we can take the cube roots of any factor separately. In other words, for any real values 𝑎 and 𝑏, the cube root of 𝑎 times 𝑏 is equal to the cube root of 𝑎 times the cube root of 𝑏.

We can use this property to reduce the size of the numbers inside the cube roots. For instance, we can see that 375 is equal to five cubed times three. Similarly, we can calculate that 81 is three cubed times three. This allows us to rewrite the expression we want to evaluate as the cube root of five cubed times three plus the cube root of three cubed times three all cubed. This is useful since we can take the cube root of each factor separately. And we know that the cube root of 𝑎 cubed will be equal to 𝑎. This means we can take cubic factors out of the radical by taking their cube roots.

First, we can set 𝑎 equal to five cubed and 𝑏 equal to three to get that the cube root of five cubed times three is equal to the cube root of five cubed times the cube root of three. We can follow the same process for the second term inside the parentheses. We set 𝑎 equal to three cubed and 𝑏 equal to three to get the cube root of three cubed times the cube root of three. We then need to cube the sum of these terms.

We can now simplify the expression inside the parentheses by evaluating the cube roots. We have that the cube root of five cubed is five and the cube root of three cubed is three. So, we obtain five times the cube root of three plus three times the cube root of three all cubed. We can now see that the two terms inside the parentheses share a factor of the cube root of three. So, we can take out this factor and add them together by adding five and three. This gives us eight times the cube root of three all cubed.

We now have a product raised to an exponent. So, we can now evaluate this expression using the laws of exponents. In particular, we know that for any real numbers 𝑎 and 𝑏 and integer 𝑛, 𝑎 times 𝑏 all raised to the power of 𝑛 is equal to 𝑎 raised to the power of 𝑛 times 𝑏 raised to the power of 𝑛 provided that these are all well defined. We can apply this result with 𝑎 equal to eight, 𝑏 equal to the cube root of three, and 𝑛 equal to three to get eight cubed times the cube root of three cubed.

Finally, we can evaluate. We calculate that eight cubed is 512 and the cube root of three cubed is three, giving us 512 times three, which we can calculate is 1,536.

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