Video: Computing Numerical Expressions Involving Square Roots

Simplify 5⁷ + (√5)¹⁴ + (√5)¹⁴ + (√5)¹⁴ + (√5)¹⁴.

02:38

Video Transcript

Simplify five to the seventh power plus the square root of five to the 14th power plus the square root of five to the 14th power plus the square root of five to the 14th power plus the square root of five to the 14th power.

The first thing that I wanna do is rewrite all of these square roots as five to the one-half power. The square root of five is the same thing as saying five to the one-half power. And we’re still taking this five to the one-half power to the 14th power in all four cases.

And then, we’ll need to consider the power rule. 𝑥 to the 𝑎 power to the 𝑏 power is the same thing as saying 𝑥 to the 𝑎 times 𝑏 power. And that means five to the one-half power to the 14th power equals five to the one-half times 14 power. And that would be true in all four of these cases.

We can multiply 14 times one-half, which equals seven. We’re dealing with five to the seventh power. All four of these simplify to five to the seventh power. And we can’t forget the other five to the seventh power term we started with.

What’s happening now is we’re adding five to the seventh power to itself five times. Let’s consider what 𝑥 to the 𝑎 power plus 𝑥 to the 𝑎 power would be. It would be two times 𝑥 to the 𝑎 power. We have two 𝑥 to the 𝑎 power terms. In our case, we have five five to the seventh power terms, five times five to the seventh power, which we can rewrite like this: five times five to the seventh power.

We’ll need another rule. This is for multiplying exponents with the same base and different powers. 𝑥 to the 𝑎 power times 𝑥 to the 𝑏 power equals 𝑥 to the 𝑎 plus 𝑏 power. And we have the whole number five, which we could write as five to the first power.

In that case, we would add one plus seven. We would add the exponents and find that this expression simplifies to five to the eighth power.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.