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Simplify 3a times a to the fifth times a squared.

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So the exponent property we can use here is

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if we have the same base, in this case it's a,

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if we have it raised to the x power and we're

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multiplying it by a to the y power,

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then this is just going to be equal to a to the x plus y power,

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and we'll think about why that works in a second,

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so let's just apply it here, and let's start with

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the a to the fifth times a squared. So if

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we just apply this property over here, this will result

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in a to the five plus (2nd) power.

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so that's what those guys reduce to or simplify to

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and of course we still have, we still have the

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3a out front.

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Now, what I want to do to take a little bit to the side

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and realize why this works.

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Let's think about a to the fifth times a squared

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means. A to the fifth literally means

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a times a times a times a times a.

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Now, a squared literally means a times a.

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And we're multiplying the two times each other.

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So we're multiplying these five a's times these two a's,

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so what have we just done?

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We're multiplying a by itself five times, then

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another two times, so let me make it clear

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this over here is a to the fifth, this over here

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is a squared, when you multiply the two

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you're multiplying a by itself seven times! Five plus two!

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So this is a to the seventh power. (A to the five plus two power.)

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So this simplifies to 3a times a to the seventh power.

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Now, you might say, how do I apply the property

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over here? What is the exponent on the a?

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And remember, if I just have an a over here,

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this is equivalent to a to the first power.

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So I can rewrite 3a as three times a to the first power

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And now maybe it makes it a little bit more clearer

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a to the first power, and the association property

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of multiplication-- I can do the multiplication of

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the a's first before I worry about the 3's!

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So I can multiply these two guys first.

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So a to the first times a to the seventh,

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I just have to add the exponents cause I have the same base and I'm taking the product,

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that's going to be a to the eighth power,

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and I still have this three out front.

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So, 3a times a to the fifth times a squared simplifies to

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3a to the eighth power.