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09 Mar 2022, 04:15
Kudos
Bookmarks
C
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:
79% (01:48) correct
21%
(02:19)
wrong
based on 140
sessions
History
Date
Time
Result
Not Attempted Yet
For integers x and y, \(25^{(−10)}*100^x=2^y\). What is the value of y?
A. -10
B. 10
C. 20
D. 25
E. 40
A. -10
B. 10
C. 20
D. 25
E. 40
09 Mar 2022, 10:35
Kudos
Bookmarks
Bunuel
Nice question!
STRATEGY: When it comes to solving equations with variables in the exponents, it's usually best to rewrite the powers with the same bases.
Useful properties: \((x^a)^n = x^{an}\) and \((x^ay^b)^n = x^{an}y^{bn}\)
Rewrite \(25\) and \(100\) as follows: \((5^2)^{(−10)}(5^2 2^2)^x=2^y\)
Apply the above properties to get: \((5^{(−20)})(5^{2x})(2^{2x})=2^y\)
Use the Product Law to combine the two powers with base \(5\) to get: \((5^{(−20 + 2x)})(2^{2x})=2^y\)
Important: Since the right side of the equation doesn't have any powers of \(5\), it must be the case that \(5^{(−20 + 2x)}\) evaluates to be \(1\) (aka \(5^0\))
In other words: \(−20 + 2x = 0\)
Solve to get: \(x = 10\), which means our simplified equation becomes: \((5^{(−20 + 20)})(2^{20})=2^y\), which simplifies to be \(2^{20}=2^y\), which means \(y = 20\)
Answer: C
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