Question Video: Evaluating an Expression with Integer Exponents Mathematics

Video Transcript

Find the value of four squared plus eight all over two squared in the simplest form.

To find the value of the given expression in the simplest form, there are a couple of different approaches we could take. One way is to write out the powers in expanded form and evaluate from there, where we recall that for a nonzero base 𝑎 and positive integer power or exponent 𝑛, 𝑎 raised to the 𝑛th power is equal to 𝑛 instances of 𝑎 multiplied together.

In our expression, we have two numbers, which are the bases, raised to the power two: in other words, where the exponent 𝑛 is equal to two and we have a base 𝑎 equals four in the numerator and a base of two in the denominator. We can write each of these in expanded form so that we have four times itself in the numerator and two times itself in the denominator. Evaluating our two expansions gives 16 plus eight in the numerator and four in the denominator. That’s 24 over four, which then evaluates to six.

Hence, the value of four squared plus eight over two squared is equal to six. We could’ve found this in a different way by looking for common factors in the numerator and denominator and canceling. So, let’s see how this might work.

Noticing first that four equals two squared and next that eight is equal to two times two squared, in our numerator we can replace four squared with two squared all squared and eight with two times two squared. And we see we have a common factor of two squared which we can take outside some parentheses to give a numerator of two squared multiplied by two squared plus two. We can then divide our numerator and denominator by two squared, and we’re left with two squared plus two. That’s two times two plus two, which is four plus two. And as expected, evaluating this, we find the value of the given expression in its simplest form is six.