Without calculator, I have to determine which of the following is larger:
$2^{350}$ or $5^{150}$
I know only very basic exponential laws, and haven't covered logarithms yet. Tried various algebraic simplification methods but had no luck.
Any help is much appreciated, thanks in advance.
$\endgroup$ 13 Answers
$\begingroup$Hint
This should help:
$$2^7=128>125=5^3$$
$\endgroup$ $\begingroup$To evaluate these values, they must be placed on the same “platform”.
Find $x$ such that $2^{350} = (x)^{50}$. Similarly, find $y$ such that $5^{150} = (y)^{50}$.
These two numbers are now raised to the same platform and therefore can be compared (now).
If $x > y$ then..., otherwise ......
$\endgroup$ $\begingroup$The greatest common divisor between $350$ and $150$ is $50$ and we have
$350 = 7 \cdot 50$
and
$150 = 3 \cdot 50$
There's an exponent power rule that states $a^{x \cdot y} = (a^x)^y$
Thus you're comparing
$2^{7 \cdot 50}$ or $(2^7)^{50}$ or $128^{50}$
and
$5^{3 \cdot 50}$ or $(5^3)^{50}$ or $75^{50}$
Since $128 > 75$, then $128^{50} > 75^{50}$
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