### Video Transcript

Which of the following is equivalent to four π to the fourth power plus 20π squared π squared plus 25π to the fourth Power? A) Four π squared plus 25π squared squared, B) four π plus 25π to the fourth power, C) two π squared plus five π squared squared, or D) two π plus five π to the fourth power.

If we look closely at this expression, we could rewrite four π to the fourth power as two π squared squared. And we could rewrite 25π to the fourth power as five π squared squared. But what about the middle? I know that 20 equals two times two times five. And so Iβm starting to see a pattern. We now have something that says two π squared squared plus two times to π squared times five π squared plus five π squared squared. This is good because we recognize the pattern. π squared plus two ππ plus π squared is equal to π plus π squared. In our case, the capital π΄ would be equal to two π squared. And the capital π΅ would be equal to five π squared. And so we could say that our π΄ plus π΅ squared would be two π squared plus five π squared squared, which is option C.

Solving using this method relies upon you recognizing patterns and the expression we were given. But what if you didnβt recognize that or at least didnβt initially recognize that? Letβs take this expression and then consider the four answer choices using a square. Option A says four a squared plus 25π squared squared. And so we put four π squared and 25π squared on the left. And on the top of our square, in the top left hand corner, weβll multiply four π squared times four π squared, which gives us 16π to the fourth. This is more than our original expression. And so we know that A will not work. In option B, we have four π plus 25π. But weβre taking it to the fourth power. This will be equal to four π plus 25π times itself four times. This would result in a far greater value than the one weβve been given.

Letβs consider option C one more time. Itβs two π squared plus five π squared times two π squared plus five π squared. And we get four π to the fourth power, 10 π squared π squared, two times five is 10 and π squared times π squared is just π squared π squared. We get the same thing in the bottom left, 10 π squared π squared. And then we have 25π to the fourth power. We recognize our four π to the fourth power and our 25π to the fourth power. 10π squared π squared and 10π squared π squared are like terms and can be combined together to equal 20π squared π squared. And this confirms that C), two π squared plus five π squared squared, is equal to four π to the fourth power plus 20π squared π squared plus 25π to the fourth power.

We would have two π plus five π times itself four times. Using the square method, you would multiply two π plus five π times itself. And then you would multiply whatever you found by itself again. And you would end up with too many terms for it to be equivalent to the expression we started with. And so our final answer is two π squared plus five π squared squared.