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16 Sep 2014, 01:51
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
81% (00:47) correct
19%
(01:40)
wrong
based on 32
sessions
History
Date
Time
Result
Not Attempted Yet
\(x^8 - y^8 =\)
A. \((x^4 - y^4)^2\)
B. \((x^4 + y^4)(x^2 + y^2)(x + y)(x - y)\)
C. \((x^6 + y^2)(x^2 - y^6)\)
D. \((x^4 - y^4)(x^2 - y^2)(x - y)(x + y)\)
E. \((x^2 - y^2)^4\)
A. \((x^4 - y^4)^2\)
B. \((x^4 + y^4)(x^2 + y^2)(x + y)(x - y)\)
C. \((x^6 + y^2)(x^2 - y^6)\)
D. \((x^4 - y^4)(x^2 - y^2)(x - y)(x + y)\)
E. \((x^2 - y^2)^4\)
16 Sep 2014, 01:51
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Official Solution:
\(x^8 - y^8 =\)
A. \((x^4 - y^4)^2\)
B. \((x^4 + y^4)(x^2 + y^2)(x + y)(x - y)\)
C. \((x^6 + y^2)(x^2 - y^6)\)
D. \((x^4 - y^4)(x^2 - y^2)(x - y)(x + y)\)
E. \((x^2 - y^2)^4\)
You need to recognize the expression as a difference of squares. Like all other even powers, \(x^8\) is a square, equal to \((x^4)^2\), so we use the template \(a^2 - b^2 = (a + b)(a - b),\) with \(a = x^4\) and \(b = y^4\):
\(x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^4 - y^4)\)
We continue breaking down the second part of the resulting expression, which is also a difference of squares.
\(x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2)\)
And we’re not done yet, because the last expression is of course also a difference of squares!
\(x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2) = (x^4 + y^4)(x^2 + y^2)(x + y)(x - y)\)
This final product matches the expression in choice (B), so the correct answer is (B).
Plugging numbers is probably too time-consuming in this case. Among positive integers, only 0 and 1 are easy to compute the eighth power of (unless you've memorized that \(2^8 = 256\)). Moreover, several of the answer choices are designed to give you 0 if you choose \(x = y = 1\).
If you did plug in \(x = 2\) and \(y = 1\), then you would get the following for choice (B):
\(256 - 1 = 255 = (16 + 1)(4 + 1)(2 + 1)(2 - 1) = (17)(5)(3)\)
If you happen to know already that \(2^8 = 256\), then you could get 255 as your target number relatively quickly.
Answer: B
\(x^8 - y^8 =\)
A. \((x^4 - y^4)^2\)
B. \((x^4 + y^4)(x^2 + y^2)(x + y)(x - y)\)
C. \((x^6 + y^2)(x^2 - y^6)\)
D. \((x^4 - y^4)(x^2 - y^2)(x - y)(x + y)\)
E. \((x^2 - y^2)^4\)
You need to recognize the expression as a difference of squares. Like all other even powers, \(x^8\) is a square, equal to \((x^4)^2\), so we use the template \(a^2 - b^2 = (a + b)(a - b),\) with \(a = x^4\) and \(b = y^4\):
\(x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^4 - y^4)\)
We continue breaking down the second part of the resulting expression, which is also a difference of squares.
\(x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2)\)
And we’re not done yet, because the last expression is of course also a difference of squares!
\(x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2) = (x^4 + y^4)(x^2 + y^2)(x + y)(x - y)\)
This final product matches the expression in choice (B), so the correct answer is (B).
Plugging numbers is probably too time-consuming in this case. Among positive integers, only 0 and 1 are easy to compute the eighth power of (unless you've memorized that \(2^8 = 256\)). Moreover, several of the answer choices are designed to give you 0 if you choose \(x = y = 1\).
If you did plug in \(x = 2\) and \(y = 1\), then you would get the following for choice (B):
\(256 - 1 = 255 = (16 + 1)(4 + 1)(2 + 1)(2 - 1) = (17)(5)(3)\)
If you happen to know already that \(2^8 = 256\), then you could get 255 as your target number relatively quickly.
Answer: B
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