Evaluate each radical expression if it is a real number. If...

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Key Concepts

Real Numbers versus Complex Numbers

This concept distinguishes numbers on the real number line from numbers that have both real and imaginary parts. For radical expressions, if the radicand is negative, the square root is not a real number and is instead associated with complex number representations.

Negation of Radical Expressions

The negation of a radical expression involves applying a negative sign to the result of the square root. This operation can change the sign of the value obtained but does not alter the domain restrictions of the radical itself.

Square Root

A square root represents the nonnegative number which, when multiplied by itself, equals the original number. In the context of real numbers, the square root operation is defined for nonnegative values, making it a function that returns only the principal (nonnegative) square root.

Radicand

The term under the radical sign is called the radicand. Its value is critical in determining whether the radical expression is real or needs to be considered within the complex numbers; nonnegative radicands yield real results, while negative ones do not in standard real arithmetic.

Transcript

00:01 Hey guys, today i'm solving problem number 20 on page 29 of the textbook.

00:07 This is a three -part problem.

00:10 In the first part, we are asked to figure out what the square root of 64 is.

00:14 In the second part, we are asked to figure out what the third root of negative 64 is.

00:19 And in part c, we are asked to figure out what the fifth root of negative 32 is.

00:24 Now, in part a, the square root of 64 is the same thing as the second fruit.

00:31 The 64.

00:32 So if you want, you can write it like that.

00:36 And basically what this is asking is this is indicative how many times some number will call it x.

00:45 It needs to be multiplied by itself in order to yield 64.

00:53 So i repeat that again.

00:55 This is basically saying x times x or x squared must equal 64.

01:08 So what number times itself equals 64? so what number times itself equals 64? well, that means that x must be 8 because 8 times 8 equals 64.

01:22 So that's our answer for part a.

01:24 Now in part b, it's a little bit more complicated, but similar concept.

01:29 We have a negative 64, and we have to yield that negative 64 by multiplying some number.

01:36 We'll call it y for this one, three times.

01:41 Now remember, the negative is there.

01:44 So a negative times a negative times a negative.

01:47 Will always yield a negative number...

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