Question Video: Evaluating an Expression Involving the Product and Exponentiation of Rational Numbers | Nagwa Question Video: Evaluating an Expression Involving the Product and Exponentiation of Rational Numbers | Nagwa

Question Video: Evaluating an Expression Involving the Product and Exponentiation of Rational Numbers Mathematics • First Year of Preparatory School

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Calculate (−2/3)³ × (−9/8).

02:41

Video Transcript

Calculate negative two-thirds cubed times negative nine-eighths.

In this question, we are asked to calculate an expression involving the cube and product of a rational number. In the order of operations, we evaluate the exponent first. So let’s begin by recalling how we evaluate a rational number raised to a positive integer exponent.

We can recall that if 𝑎 over 𝑏 is a rational number and we raise this to the power of 𝑛 for a positive integer 𝑛, then we can raise the numerator and denominator to the power of 𝑛 separately. To apply this result, we first need to recall that we can rewrite negative two-thirds as negative two over three. And we will also rewrite negative nine-eighths as negative nine over eight. We can now take the cube of the numerator and denominator separately to obtain negative two cubed over three cubed multiplied by negative nine over eight.

We now need to evaluate the cubes. We can do this by recalling that cubing a number means a product of three lots of that number. So negative two cubed is negative eight and three cubed is 27. We now have the product of two fractions. We can recall that we can multiply fractions by multiplying their numerators and denominators. So 𝑎 over 𝑏 times 𝑐 over 𝑑 is equal to 𝑎 times 𝑐 over 𝑏 times 𝑑. This gives us negative eight times negative nine over 27 times eight.

We could now evaluate these products. However, it is easier to cancel the shared factors first. We can cancel the shared factor of eight in the numerator and denominator. It is worth noting that the numerator will then be negative one times negative nine. We can also cancel the shared factor of nine in the numerator and denominator. This gives us negative one times negative one over three times one. Finally, we can evaluate the products. We have negative one times negative one is one and three times one is three. So we are left with one-third, which is our final answer.

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