In Exercises 55–68, multiply using one of the rules for the squar...

Question

In Exercises 55–68, multiply using one of the rules for the square of a binomial.

(aⁿ + 4bⁿ)²

Accepted Answer

Hello, everyone. We've been asked to perform multiplication on the given expression. We were given open parentheses. Five X raised to the NTH power plus Y raised to the NTH power closed parentheses squared. Our answer choices are a 25 X raise to the two N power plus five X raise to the two N power Y raised to the two N power plus Y raise to the two N power B X raised to the two N power plus 10 X raised to the two N power Y raised to the two N power plus Y raise to the two N power C X raised to the two N power plus five X raised to the end power Y raise to the end power plus Y raise to the two N power D 25 X raised the two N power plus X raised the end power, Y raised the end power plus Y raised the two N power rewriting our original expression. I have parentheses. Five XX raised the N power plus Y raised the N power closed parentheses squared. So I have a binomial that is squared. I recall that if I wanted to, I could still rewrite this as two binomials and use foil. However, there is a shortcut, the format to the shortcut. I'm going to write off to the right hand side. And it's that if you have in parentheses two terms, so A plus B squared, our trinomial that results from this is A squared plus two A B plus B squared. What this means for us if we compare this to the expression, we're given A is the same as five X raised to the nth power in our example. And B is the same as Y raised to the nth power. So we're gonna use this format to find our trinomial. So the first term in our trinomial is A squared. So we said that R A is five X raised to the nth power. So if I wanted to put that in parentheses, five X raised the nth power close parentheses squared, I'm gonna multiply that out. After I set this up, my middle term is two A B. So this is two multiplied by A multiplied by B. So for us, this would be two multiplied by five X rays to the end power multiplied by Y raised the end power. And our last term comes from B squared. So why raise to the end power squared? And I put that in parentheses. So right now I have in parentheses, five X raised to the end power, the whole parentheses squared plus two multiplied by five X raised to the end power multiplied by Y raise to the end power plus in parentheses. Why raise the end power? The parentheses squared? So now I'm gonna clean this up. I recall that when I have a product raised to an exponent, I multiply that exponent on both of the bases. So I'm gonna use the two as a square on both the base of five and the base of X. So five squared is X raised the nth power raised to the second power. When we raise a power to a power, we multiply our exponents. So we'll get X raised the two end power as our first term for a middle term, I'm gonna multiply the coefficients first. So I have two times five, which is 10 and that's also multiplied by X raise to the N power Y raised to the nth power because they are not like bases. I cannot combine them. They just sit next to each other. Our last term is why raise to the NTH power squared. And again, when we raise a power to a power, we keep our base and we multiply our exponents. So we get, why raise to the two nth power? So our product here is 25 X raised the two nth power plus 10 X raised the nth power. Why raised the nth power plus why raised the two nth power? This is our answer and it's answer choice. D have a nice day.

In Exercises 55–68, multiply using one of the rules for the square of a binomial. (aⁿ + 4bⁿ)²

Key Concepts

Binomial Expansion

Square of a Binomial

Exponents and Powers

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