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Which of the following is equivalent to four π to the fourth power plus 20π squared π squared plus 25π to the fourth Power?
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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.
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If we look closely at this expression, we could rewrite four π to the fourth power as two π squared squared.
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And we could rewrite 25π to the fourth power as five π squared squared.
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But what about the middle?
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I know that 20 equals two times two times five.
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And so Iβm starting to see a pattern.
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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.
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This is good because we recognize the pattern.
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π squared plus two ππ plus π squared is equal to π plus π squared.
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In our case, the capital π΄ would be equal to two π squared.
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And the capital π΅ would be equal to five π squared.
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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.
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Solving using this method relies upon you recognizing patterns and the expression we were given.
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But what if you didnβt recognize that or at least didnβt initially recognize that?
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Letβs take this expression and then consider the four answer choices using a square.
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Option A says four a squared plus 25π squared squared.
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And so we put four π squared and 25π squared on the left.
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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.
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This is more than our original expression.
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And so we know that A will not work.
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In option B, we have four π plus 25π.
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But weβre taking it to the fourth power.
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This will be equal to four π plus 25π times itself four times.
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This would result in a far greater value than the one weβve been given.
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Letβs consider option C one more time.
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Itβs two π squared plus five π squared times two π squared plus five π squared.
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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.
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We get the same thing in the bottom left, 10 π squared π squared.
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And then we have 25π to the fourth power.
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We recognize our four π to the fourth power and our 25π to the fourth power.
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10π squared π squared and 10π squared π squared are like terms and can be combined together to equal 20π squared π squared.
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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.
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We would have two π plus five π times itself four times.
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Using the square method, you would multiply two π plus five π times itself.
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And then you would multiply whatever you found by itself again.
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And you would end up with too many terms for it to be equivalent to the expression we started with.
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And so our final answer is two π squared plus five π squared squared.