Comparing large exponents

Question

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.

Answer 1

Hint

This should help:

$$2^7=128>125=5^3$$

Answer 2

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 ......

Answer 3

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 $128^{50} > 75^{50}$

Answer 4

There is a simpler way which does not require the gcd or "common base" and works independent if the exponents share some bases:

If you have the two numbers:

$2^{350}$ and $5^{150}$

Choose the smaller of the two exponents, i.e. 150 and apply the reverse power to both sides, i.e.

${2^{350}}^{1/150}$ and ${5^{150}}^{1/150}$

This can be simplified to

$2^{350/150}$ and $5^1$

and further to

$2^{2.333...}$ and $5^1$

which makes the two sides easy to compare.

This way you can compare large numbers with huge exponents easily as you always get one of the two numbers to have exponent "1".

Note: You may need to compute non-integer exponents for the remaining power.