Which number is larger? $4^{25}$ or $9^{15}$. Why? I know that it used powers of 2 and 3 but how?
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1$\begingroup$ We can do this in our heads. Simplify to a power of 2 vs. a power of 3, then take out (taking a root) the greatest common divisor of the exponents. When you have done that, you'll have only a couple of two digit numbers to compare. $\endgroup$– hardmathJul 17, 2014 at 19:43
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4 Answers
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$4^{25} = 2^{50} = 32^{10} > 27^{10} = 3^{30} = 9^{15}$ So first one larger.
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We want to compare $4^{25}$ and $9^{15}$. Rewrite as $2^{50}$ and $3^{30}$. Now, we have that
$$
\frac{2^{50}}{3^{30}}>1\iff \frac{2^5}{3^3}>1 \iff \frac{32}{27}>1
$$
and since the latter is true, we have $2^{50}>3^{30}$.
Note that the first equivalency holds because $\sqrt[10]{\cdot}$ is a monotone function.
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It's not hard to see that $$4^5=1024>9^3=729$$ and now raise to the power $5$.
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1$\begingroup$ Actually $4^5 = 1024$ and $9^3 = 729$ (though of course this doesn't affect the conclusion). $\endgroup$– David ZJul 18, 2014 at 3:10
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$\begingroup$ LoL it's an incomprehensible error I do not know how I did it. Thanks. $\endgroup$– user63181Jul 18, 2014 at 5:40
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$log(4^{25})=50log2=50(.3)=15 \\$(approximately) $$ \\ $$ $log(9^{15})=30log3=14.31 $(approximately)
So, 1st one is bigger.
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Alternatively, $(4^{25})=(4^5)^5$ and $(9^{15})=(9^3)^5$ $$ \\ $$ $4^5>9^3$
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$\begingroup$ You've made a mistake in the 2nd approximation anyway. $\endgroup$– hardmathJul 17, 2014 at 19:44
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$\begingroup$ Also, I think you meant $\log{9^{15}} = 30 \log{3}$ not $45 \log{3}$ $\endgroup$ Jul 17, 2014 at 19:44