Algebra Frenzy - Part 3

Algebra Level pending

If 2 a = 5 b = 10 \displaystyle 2^a=5^b=10 , 1 a + 1 b = ? \displaystyle \frac{1}{a}+\frac{1}{b}=?


The answer is 1.

Alice Smith by Alice Smith , 20, China

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1 solution

Chris Lewis
May 18, 2019

Rewriting the first equations, we find 2 = 1 0 1 a 2=10^{\frac{1}{a}} and 5 = 1 0 1 b 5=10^{\frac{1}{b}} .

Multiplying these together gives

1 0 1 = 1 0 1 a + 1 b 10^1=10^{\frac{1}{a}+\frac{1}{b}}

and comparing exponents we see 1 a + 1 b = 1 \frac{1}{a}+\frac{1}{b}=\boxed1 .

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