In Exercises 69–82, multiply using the rule for the product of th...

Question

In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.

(5aⁿ − 7)(5aⁿ + 7)

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is nine times K to the power of N minus 11 times nine times K to the power of N plus 11. Now, there's two approaches that we can take to this, we can either foil the two quantities together or we can go ahead and use the difference of squares to simplify this expression. Let's go ahead and use the difference of squares to simplify the expression. Now, the difference of square states that if you have the binomial in the form of A squared minus B squared, then this can be factored to be A plus B Times A. -7. And we're going to be using this property going from right to left. See here A plus B times A minus B matches the form of the expression given to us. Here, we can go ahead and label A to be nine times K to the power of N and we can label B to be 11. So if we plug this into the binomial of A squared minus B squared, we can rewrite the expression as nine times K to the power of N squared minus 11 squared. Now, in order to simplify this, we're going to use our exponential distribution. And on the left hand side, we're going to multiply N and two together and we're going to distribute the exponent of 2-9. So the left hand side can be rewritten as nine squared times K to the power of two times N, which is going to give us two N. And on the right hand side, 11 squared is going to give us 121. So we're going to be subtracting 121 from the first part. Now, one more simplification we can do is to evaluate nine squared, nine squared is the same thing as nine times nine. So that's going to give us 81 times K to the power of two N minus 121. And this is going to be the simplified form of the given expression. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms. (5aⁿ − 7)(5aⁿ + 7)

Key Concepts

Product of Sum and Difference

Algebraic Expressions

Exponent Rules

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