domu904 wrote:
If \((\frac{1}{5})^m * (\frac{1}{4})^{18} = \frac{1}{2*(10)^{35}}\), then m = ?
A. 17
B. 18
C. 34
D. 35
E. 36
Exponent property #1: \((\frac{a}{b})^k=\frac{a^k}{b^k}\)
Exponent property #2: \((b^x)^y = b^{xy}\)
Exponent property #3: \((ab)^x = a^xb^x\)------------------------------------------------------
Given: \((\frac{1}{5})^m * (\frac{1}{4})^{18} = \frac{1}{2*(10)^{35}}\)
Applying exponent property#1, we get: Given: \((\frac{1^m}{5^m})(\frac{1^{18}}{4^{18}}) = \frac{1}{2*(10)^{35}}\)
Simplify to get: Given: \((\frac{1}{5^m})(\frac{1}{4^{18}}) = \frac{1}{2*(10)^{35}}\)
ASIDE: To determine the value of m, we must rewrite both sides of the equation with similar base.
So, for the left side, we'll rewrite \(4\) as \(2^2\)
For the right side, we'll rewrite \(10\) as \(2 \times 5\)
We get: \((\frac{1}{5^m})(\frac{1}{(2^2)^{18}}) = \frac{1}{2*(2 \times 5)^{35}}\)
Applying exponent property#2, we get: \((\frac{1}{5^m})(\frac{1}{(2^{36}}) = \frac{1}{2*(2 \times 5)^{35}}\)
Applying exponent property#3, we get: \((\frac{1}{5^m})(\frac{1}{(2^{36}}) = \frac{1}{2*(2^{35})(5^{35})}\)
Since \(2*(2^{35} = 2^{36}\), we can write: \((\frac{1}{5^m})(\frac{1}{(2^{36}}) = \frac{1}{(2^{36})(5^{35})}\)
NOTE: Notice that I really didn't need to spend so much time working on the powers of 2, since the variable m was the exponent of 5.
Had I focused solely on the powers of 5, I could have answered the question MUCH faster. That said, I wanted to show all of the exponent properties at work.
Cheers,
Brent
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