Notice that we can rewrite the second term in terms of the base of four.  Sixteen is equal to four squared.

$\displaystyle 16= 4^2$

We can replace this term with 16 in order to multiply, and then add the exponents.

$\displaystyle 4^{30}\cdot 16^{12} =4^{30}\cdot 4^{2(12)} =4^{30}\cdot 4^{24}$

According to the rule of exponents, whenever powers of a similar base are multiplied, the exponents can be added.

$\displaystyle x^A \cdot x^B = x^{( A + B )}$

The answer is:  $\displaystyle 4^{54}$

When dividing powers of a similar base, the exponents can be subtracted.

$\displaystyle \frac{4^{34}}{4^{36}} = 4^{34-36}=4^{-2}$

Simplify the negative exponent.

$\displaystyle x^{-n} = \frac{1}{x^n}$

$\displaystyle 4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

The answer is:  $\displaystyle \frac{1}{16}$

In order to simplify this, we will need to rewrite the base 27 as a common base of three in order to be able to subtract the exponents.

$\displaystyle 27 = 3^3$

Rewrite the fraction.

$\displaystyle \frac{3^{15}}{27^{18}} = \frac{3^{15}}{3^{3(18)}} = \frac{3^{15}}{3^{54}}$

Subtract the exponents.

$\displaystyle 3^{15-54} = 3^{-39}=\frac{1}{3^{39}}$

The answer is:  $\displaystyle \frac{1}{3^{39}}$

In order to subtract the exponents, we will need similar bases.  

Rewrite the nine as a base of three.  Nine is equivalent to $\displaystyle 3^2$.

The expression becomes:  

$\displaystyle \frac{3^{260}}{9^{132}}= \frac{3^{260}}{3^{2(132)}}=\frac{3^{260}}{3^{264}}$

Now that the bases are equal, we can subtract the exponents and simplify.

$\displaystyle 3^{260-264} = 3^{-4}=\frac{1}{3^4} = \frac{1}{81}$

The answer is:  $\displaystyle \frac{1}{81}$

In order to simplify this, we will need to convert the base eight to base two.

$\displaystyle 8=2^3$

Rewrite the second term.

$\displaystyle 2^{18}\cdot 8^{500} = 2^{18}\cdot (2^3)^{500} = 2^{18}\cdot2^{1500}$

Because the bases are now similar, we can add the exponents.

$\displaystyle 2^{18}\cdot2^{1500} = 2^{1500+18} = 2^{1518}$

The answer is:  $\displaystyle 2^{1518}$

Rewrite the numerator and denominator using the base power of two.

$\displaystyle \frac{8^{100}}{16^{350}} = \frac{2^{3(100)}}{2^{4(350)}}$

Use the product rule of exponents to simplify the powers.

$\displaystyle \frac{2^{300}}{2^{1400}}$

Now that the bases are similar, the exponents on the numerator and denominator can be subtracted.

$\displaystyle \frac{2^{300}}{2^{1400}} = 2^{300-1400} = 2^{-1100} = \frac{1}{2^{1100}}$

The answer is:  $\displaystyle \frac{1}{2^{1100}}$

Use the power rule of exponents to simplify.  Multiply the power of 42 with the power of two outside the parentheses.

$\displaystyle (2^{42})^2\cdot 2^{24} = 2^{84}\cdot 2^{24}$

When the powers of the same bases are multiplied, the powers can be added.

$\displaystyle 2^{84+24} = 2^{108}$

The answer is:  $\displaystyle 2^{108}$

In order to be able to subtract the exponents, we will need to have similar bases in the numerator and denominator.

$\displaystyle 8=2^3$

$\displaystyle 4=2^2$

Replace the values and simplify.

$\displaystyle \frac{8^{200}}{4^{100}} = \frac{2^{3(200)}}{2^{2(100)}} = \frac{2^{600}}{2^{200}}$

Subtract the exponents.

The answer is:  $\displaystyle 2^{400}$

Change the base of the numerator so that it matches the denominator.

$\displaystyle 25= 5^2$

We can rewrite the numerator as five squared.

$\displaystyle \frac{25^{30}}{5^{90}} = \frac{(5^2)^{30}}{5^{90}}$

Use the product rule of exponents to simplify the numerator.

$\displaystyle \frac{(5^2)^{30}}{5^{90}} = \frac{5^{60}}{5^{90}}$

Subtract the powers.

$\displaystyle 5^{60-90} = 5^{-30}$

Rewrite the negative exponent as a fraction.

The answer is:  $\displaystyle \frac{1}{5^{30}}$

Rewrite each term as a base of five.

$\displaystyle 125 = 5^3$

$\displaystyle 5=5^1$

$\displaystyle 25=5^2$

Replace the terms.

$\displaystyle 125(5^{50})(25^{100}) = 5^3(5^{1(50)})(5^{2(100)})= (5^3)(5^{50})(5^{200})$

Now that the similar bases with exponents are multiplied, the powers can be added.

$\displaystyle (5^3)(5^{50})(5^{200}) = (5^{3+50+200}) = 5^{253}$

The answer is:  $\displaystyle 5^{253}$