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Simplify seven π₯ to the fourth power times eight π₯ to the seventh power.
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There are two things we know about multiplication.
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It is associative and commutative.
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The associative property tells us that the product will be the same no matter how the numbers are grouped.
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And the commutative property tells us that the order in which we perform the calculation really doesnβt matter.
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And so, weβre going to begin by sort of unsimplifying each term in our expression.
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When we do, we get seven times π₯ to the fourth power times eight times π₯ to the seventh power.
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Weβre now going to switch around π₯ to the fourth power and eight.
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And so, we see that our expression now becomes seven times eight times π₯ to the fourth power times π₯ to the seventh power.
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And in fact, we can work out the numerical part.
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We know that seven times eight is 56, which means that this becomes 56 times π₯ to the fourth power times π₯ to the seventh power.
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But what do we do with these algebraic parts?
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Well, we have a rule for multiplying exponential terms.
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As long as the base is the same, we add the exponents.
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So, π₯ to the power of π times π₯ to the power of π is π₯ to the power of π plus π.
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This in turn means that π₯ to the fourth power times π₯ to the seventh power is π₯ to the power of four plus seven.
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And since four plus seven is 11, this becomes π₯ to the 11th power.
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Now, in fact, we know that we donβt really want to include that multiplication symbol.
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And so, we simplify this further to get 56π₯ to the 11th power.
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And so, when we simplify seven π₯ to the fourth power times eight π₯ to the seventh power, we get 56π₯ to the 11th power.