Question Video: Simplifying Algebraic Expressions Involving Exponents and Square Roots Mathematics

Video Transcript

Simplify the square root of 100𝑥 to the 16th power.

In this question, the value of 𝑛 in our 𝑛th root is omitted. When it is omitted, we assume that it’s equal to two, and that’s why we define it as being the square root. So we’re going to simplify this expression by applying some of the properties of 𝑛th roots, where 𝑛 is equal to two. The first property we apply looks at finding the product of 𝑛th roots. This tells us that when the 𝑛th root of 𝑎 and the 𝑛th root of 𝑏 are well defined in real numbers, then the 𝑛th root of 𝑎 times 𝑏 is also defined. It’s the 𝑛th root of 𝑎 times the 𝑛th root of 𝑏. We can omit this value of 𝑛 and say that equivalently the square root of 𝑎 times the square root of 𝑏 is equal to the square root of 𝑎𝑏.

We can then apply this property in reverse. And it allows us to separate the square root of 100𝑥 to the 16th power and write it as the square root of 100 times the square root of 𝑥 to the 16th power. And of course, if we know our square numbers by heart, the first part of this expression is quite easy to evaluate. The square root of 100 is simply equal to 10. But what do we do with the square root of 𝑥 to the 16th power? Well, we’re going to use the rule that applies when 𝑛 is an even integer and 𝑎 is a real number. This is that the 𝑛th root of 𝑎 to the 𝑛th power is equal to the absolute value of 𝑎.

To be able to apply this rule, we do need to do a little bit of manipulation, and that involves applying one of our exponent laws. This says that 𝑥 to the eighth power squared is equal to 𝑥 to the 16th power. So we can equivalently write the square root of 𝑥 to the 16th power as the square root of 𝑥 to the eighth power squared. And then, by the earlier property, we can say that this is actually equal to the absolute value of 𝑥 to the eighth power. We can now replace each part of our earlier expression with the values 10 and the absolute value of 𝑥 to the eighth power. And we see that we’ve simplified the square root of 100𝑥 to the 16th power as 10 times the absolute value of 𝑥 to the eighth power.

Then, let’s quickly think about what the absolute value actually tells us. It takes the input, which is here 𝑥 to the eighth power, and it makes it positive. Of course, if 𝑥 is a real number, then 𝑥 to the 𝑛th power will actually be nonnegative for even values of 𝑛. In this case, our power eight is even, so we can say that 𝑥 to the eighth power will be nonnegative. It will be greater than or equal to zero if 𝑥 is a real number. This means the absolute value symbols are actually unnecessary here, and so we can further simplify our expression. When we do, we find that the square root of 100𝑥 to the 16th power can be simplified to 10 times 𝑥 to the eighth power.