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Rewrite the following expressions as the product of positive exponents, and then evaluate the expression

2
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when x is equal to 2

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So let's do this first part

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They want us to rewrite it as the product of positive

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exponents, so they don't like the fact that we have this

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negative third power and the 5 to the negative second power

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so you want to rewrite those as negative exponents.

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And to do that, we just have to remember that something

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to a negative power is just (let me write it over here)

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if I have x to the negative 2 power, this is the exact

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same thing as 1 over x to the positive 2 power. If I have

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a to the negative b power, this is the exact same thing as

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one over a to the b power

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You can get rid of that negative by just putting a 1 over

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this whole thing. By taking the inverse of the whole expression

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you can get rid of that negative.

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So let's do that over here. x to the negative 3 power,

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that is equal to this (let me do this in a different

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color). x to the negative 3rd power, that's the exact

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same thing as one over x to the third power, by definition

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Now we're going to multiply that by 5 to the negative 2

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power. Well, 5 to the negative 2 power, same exact thing

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that's the same thing as 1 over 5 to the 2nd power,

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1 over 5 to the positive 2 power.

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And then finally, we have just a straight up x squared

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the exponent is already positive there, and we can just

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leave that as times x squared.

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So we did the first part; we rewrote the following

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expression as a product of positive exponents.

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Now let's evaluate the expression when x is equal

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to two. So everywhere we see an x, we should put a 2. So this

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becomes one over two to the (let me do this in the

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same color). This becomes one over two to the third

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power times 1 over 5 squared times two squared.

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Every place we saw an x, we're now putting a 2.

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Two to the third power (just as a reminder)

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that's 2 times 2 times 2. That is equal to 8

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So this is equal to one over eight times one over

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five squared.

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Five squared is equal to 5 times 5 which is equal to 25.

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This is equal to 25. And we're going to multiply that

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times two squared, so that's

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2 squared is equal to 2 times 2, which is equal

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to 4. So times 4.

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And you can even view this as times 4 over 1

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And so you multiply all the numerators

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That's what we do when we multiply fractions

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You multiply all the numerators

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One times One times four is equal to 4

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And you multiply the denominators

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8 times 25 is going to be 200

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And you can simplify this - the numerator and

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the denominator are both divisible by 4

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Divide the numerator by 4 and you get 1

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Divide the denominator by 4 and you get 50

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So this is equal to one over 50.

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And just so you see there, there's actually a way to

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simplify it before we got to this stage over here.

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Actually from this stage we could have multiplied it out

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If you (let me just rewrite this thing)

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So this product, if we multiply it out, this 2 squared

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you can view it as 2 squared over 1.

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So if we were to multiply this out, the numerator

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would have been equal to 1 times 1 times 2 squared

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2 squared is just 2 times 2, so there would have been

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2 times 2 in the numerator, we can ignore the 1,

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that doesn't change the value.

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And the denominator would have been

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2 times 2 times 2 (that's what 2 to the third power is)

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2 times 2 times 2 times 5 times 5

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That's what 5 squared is, and of course multiplying it

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by one doesn't change its value.

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And then it would have been more obvious that you can

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divide the numerator and the denominator by 2 once

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you can do it again

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and then you'd just be left with one over two

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times 5 times 5.

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And this of course down here is 50.

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This 10 times 5 which is 50 and you would have

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gotten the same result.