Evaluate the radical. ( − 5 ) 2 \sqrt{\left(-5\right)^2}

Everyone. Let's take a look at our practice problem here. We've got the square root of negative 5 squared. Now, one of the things you might be thinking of is, well, Patrick, I thought you told me that I can't have negatives inside of square roots, and I did. That's perfectly true. You can have negatives in square roots, but this one's different because I don't just have negative 5 by itself. I have negative 5 that's raised to a power. It's raised to the second power over here. So one of the things I can do in this radical is I can actually just evaluate what's inside the radical first. This just becomes the square root of what's negative twenty what's negative 5 times itself? It actually just becomes positive 25. So this thing doesn't lead to an imaginary root because it's not just negative 5 by itself, It's negative 5 squared. So you can evaluate that first, turn it to 25. And now what is this asking you? It's asking you for what is the roots of 25, and that's positive, and this actually just turns out to be positive 5. So that is the answer. Alright? So a little bit of a strange question, but you might see this, but, you know, you might see this in your homeworks and your quizzes and stuff, so it's good to know. Thanks for watching.

9. Roots, Radicals, and Complex Numbers

Introduction to Radical Expressions

Intermediate Algebra

9. Roots, Radicals, and Complex Numbers

Introduction to Radical Expressions

Struggling with Intermediate Algebra?

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Multiple Choice

Evaluate the radical.
$\sqrt{{(- 5)}^{2}}$

$2.23$

$5$

$-5$

No real solution (Imaginary)

Verified step by step guidance

Recognize the expression inside the square root: $\sqrt{\left(-5\right)^2}$. This means you are taking the square root of the square of -5.

Recall the property of exponents: squaring a number, whether positive or negative, results in a non-negative value. So, $\left(-5\right)^2 = (-5) \times (-5)$.

Calculate the square inside the radical: $(-5) \times (-5) = 25$. So the expression becomes $\sqrt{25}$.

Understand that the square root function $\sqrt{x}$ returns the principal (non-negative) square root of $x$. Therefore, $\sqrt{25}$ is the non-negative number which, when squared, gives 25.

Conclude that the value of $\sqrt{\left(-5\right)^2}$ is the positive number 5, because the square root function always returns the principal (non-negative) root.

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Evaluate the radical.(−5)2\sqrt{\left(-5\right)^2}

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Evaluate the radical.
$\sqrt{{(- 5)}^{2}}$