Question Video: Solving a Word Problem by Taking Powers of Rational Numbers Mathematics • First Year of Preparatory School

Video Transcript

Find an expression for the volume of the given cube whose side lengths are two 𝑥 over five.

We begin by recalling that the volume of a cube is given by the cube of its side length. This means that if a cube has side length 𝑙, then its volume is 𝑙 multiplied by 𝑙 multiplied by 𝑙, which is 𝑙 cubed. In this question, we are told that the side length is two 𝑥 over five. This means that its volume is given by the expression two 𝑥 over five cubed. Recalling that for any rational number 𝑎 over 𝑏 and any integer 𝑛 𝑎 over 𝑏 to the 𝑛th power is equal to 𝑎 to the 𝑛th power divided by 𝑏 to the 𝑛th power, our expression simplifies to two 𝑥 cubed over five cubed.

Next, we recall that to multiply monomials, we multiply the coefficients and add the powers of the shared variables. This means that two 𝑥 multiplied by two 𝑥 multiplied by two 𝑥 is equal to eight 𝑥 cubed. And since five cubed is 125, our expression simplifies to eight 𝑥 cubed over 125. This is an expression for the volume of a cube whose side lengths are two 𝑥 over five.