_If x=−2, then √x⁶ = x³.

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Hey, everyone in this problem, we're asked to check whether the given statement is true or false. The statement we're given is if X is equal to negative four, then the square root of X to the exponent 10 is equal to X to the exponent five. And the answer choices are a true or B false. Now let's start with the expression we were given. OK. So we were given the square root of X to the exponent 10. And let's just try to simplify this on our own and see if it matches what the statement says. OK. So we have X to the exponent 10, we have a square root. OK. So we wanna write X to the exponent 10 as something squared. So we can simplify that squared with our square root. Recall that if we have a, to the exponent M multiplied by N, our power rule states that this is equal to A, to the exponent M to the exponent N. So we can break up an exponent into an exponent of an exponent if it's multiplied to get. So here we have 10. OK? We know that 10 is equal to five multiplied by two. And so we can write our square root of X to the exponent 10 as the square root of X to the exponent five squared. OK. If we were to multiply those together, we get that original exponent of 10. Now we have the square root of something squared. I recall if we have the nth root of A to the exponent N and we have is even, then this is equal to the absolute value of A. OK. So the end through and the exponent and cancel, OK. We're left with A but we have to take the expo or the absolute value. Sorry if N is even OK. So in this case, we have the square root and we have something squared. So our end value is two, our end value is therefore even. And so this is going to be equal to the absolute value of X to the exponent five. OK? We can't forget that absolute value. Now, in our statement that we're told that we have an x value of -4. OK? If X is equal to -4, OK. That means that X is negative. If we're taking the absolute value of a negative value, OK. If X is negative, sorry, let's do one more step, then X to the exponent five is going to be negative as well. OK? A negative value raised to an odd exponent gives us a negative value. OK. So this tells us that the absolute value of X to the exponent five. If we're taking the absolute value of a negative, we have to multiply by a negative. OK. So this is gonna be equal to negative x to the exponent five. All right. So we have an absolute value of something and that something is negative, the absolute value is going to turn that into a positive. And how do we go from a negative to a positive? Well, we multiply by a negative, a negative multiplied by a negative gives us a positive. So if X to the exponent five is negative, we multiply by a negative in in order to make it positive for absolute value. So the given statement was that if X is equal to negative four, then the square root of X to the exponent 10 is equal to X to the exponent five. But what we found is that if X is equal to negative four, then the square root of X to the exponent 10 is equal to negative X to the exponent five. And so the original statement was false. Thanks everyone for watching. I hope this video helped see you in the next one.

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_If x=−2, then √x⁶ = x³.

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