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Question Video: Evaluating an Algebraic Expression Involving Cubes of Rational Numbers Mathematics

If π‘Ž = βˆ’ 2/3, 𝑏 = 1/3, and 𝑐 = βˆ’9/2, find the numerical value of π‘ŽΒ³π‘Β³π‘Β³.

03:08

Video Transcript

If π‘Ž is equal to negative two-thirds, 𝑏 is equal to one-third, and 𝑐 is equal to negative nine-halves, find the numerical value of π‘Ž cubed times 𝑏 cubed times 𝑐 cubed.

In this question, we are given three rational numbers π‘Ž, 𝑏, and 𝑐 as fractions. And we are asked to evaluate the product of the cubes of these three numbers.

To answer this question, we can start by recalling that we can cube any fraction by cubing its numerator and denominator separately. So 𝑝 over π‘ž all cubed is equal to 𝑝 cubed over π‘ž cubed. We can use this to evaluate each of the cubes in the expression. We can start by substituting the values of π‘Ž, 𝑏, and 𝑐 into the expression to obtain that π‘Ž cubed 𝑏 cubed 𝑐 cubed is equal to negative two-thirds cubed times one-third cubed times negative nine-halves cubed.

We now want to cube each of π‘Ž, 𝑏, and 𝑐. Let’s start with π‘Ž cubed. We know that negative two-thirds is the same as negative two over three. So we cube the numerator and denominator to get negative two cubed over three cubed.

We can then follow the same process for 𝑏 cubed. We have that one-third cubed is equal to one cubed over three cubed. We can follow the same process to find 𝑐 cubed. We see that negative nine-halves is equal to negative nine over two. So negative nine-halves cubed is equal to negative nine cubed over two cubed.

We could start evaluating this expression. However, it is easier to cancel shared factors in the numerator and denominator. To do this, we can use the laws of exponents to write negative two cubed as negative one cubed times two cubed and negative nine cubed as negative one cubed times three cubed times three cubed. This gives us the following expression.

We can now start canceling the shared factors in the numerator and denominator. First, we can cancel the two shared factors of three cubed in the numerator and denominator. Next, we can cancel the shared factor of two cubed in the numerator and denominator. This leaves us with a denominator of one. Finally, we can calculate that negative one cubed is negative one. So the expression simplifies to give us just one.

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