Video: Computing Numerical Expressions Involving Square Roots Using Laws of Exponents

What is the value of (−√11)⁴ × (√3)⁻²?

03:20

Video Transcript

What is the value of the negative square root of 11 to the fourth power times the square root of three to the negative two power?

Let’s look at this in two ways. First, let’s just consider the negative square root of 11 to the fourth power. What this means is the negative square root of 11 times the negative square root of 11 times the negative squared of 11 times the negative square root of 11. We also need to consider the square root of three to the negative two power. And anything to the negative power means it needs to go into the denominator.

We can rewrite this then as the negative square root of 11 times itself times itself times itself over the square root of three times the square root of three. To simplify, first, we’ll multiply the negative square root of 11 times the negative square root of 11, which is going to equal 11, because the square root of 11 times the square root of 11 is 11. And any negative times a negative is a positive.

If we do this again for the second set, we’ll see that we have 11 times 11 in the numerator. And since we have the square root of three times the square root of three in the denominator, we can simplify that to say three. 11 times 11 is 121. And so the value of this expression is 121 over three. This process works well when the exponents are very small.

Let’s consider a second way to solve this. Is there a way that we can rewrite the square root? The square root is the same thing as the one-half power. And so we could rewrite this as negative 11 to the one-half power to the fourth power. And we could rewrite the square root of three to be three to the one-half power so that we have three to the one-half power to the negative two power.

And then based on our power to a power rule, we can multiply these two exponents. Three to the one-half power to the negative two power then becomes three to the negative one power. Negative two times one-half equals negative one. We should be careful here because we do have this negative value. And that’s as if it was multiplying negative one. We could rewrite then as negative one to the fourth power times 11 squared. This square is there because one-half times four equals two. And this negative one is being taken to the fourth power because it was not part of the exponent that was being taken to the one-half power.

Now we’re ready to simplify. Negative one to the fourth power is positive one. And 11 squared is 121. We also know that three to the negative one power means that three belongs in the denominator. So we can say again the value of this expression is 121 over three.

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