Simplify expressions involving exponents, including multiplication and division of powers with the same base.

Using the rule $a^m \times a^n = a^{m+n}$:

$$10^5 \times 10^4 = 10^{5+4} = 10^9$$

The simplified expression is $\boxed{\mathbf{10^9}}$.

Using the rule $a^m \times a^n = a^{m+n}$:

$$a^3 \times a^9 = a^{3+9} = a^{12}$$

The simplified expression is $\boxed{\mathbf{a^{12}}}$.

Multiply the coefficients and the variables separately:

$$5y \times 4y^4 = (5 \times 4) \times (y^1 \times y^4)$$ $$= 20 \times y^{1+4} = 20y^5$$

The simplified expression is $\boxed{\mathbf{20y^5}}$.

Using the rule $a^m \times a^n = a^{m+n}$:

$$2^2 \times 2^4 = 2^{2+4} = 2^6$$

Calculate the value:

$$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$$

The simplified expression is $\boxed{\mathbf{64}}$.

Using the rule $a^m \div a^n = a^{m-n}$:

$$m^8 \div m^5 = m^{8-5} = m^3$$

The simplified expression is $\boxed{\mathbf{m^3}}$.

Using the rule $a^m \div a^n = a^{m-n}$ (where $c = c^1$):

$$c^7 \div c^1 = c^{7-1} = c^6$$

The simplified expression is $\boxed{\mathbf{c^6}}$.

Divide the coefficients and the variables separately:

$$\frac{24x^6}{8x^4} = \frac{24}{8} \times \frac{x^6}{x^4}$$ $$= 3 \times x^{6-4} = 3x^2$$

The simplified expression is $\boxed{\mathbf{3x^2}}$.

Divide the numerical parts and the powers of 10 separately:

$$\frac{9 \times 10^9}{3 \times 10^3} = \frac{9}{3} \times \frac{10^9}{10^3}$$ $$= 3 \times 10^{9-3} = 3 \times 10^6$$

The simplified expression is $\boxed{\mathbf{3 \times 10^6}}$.

Using the rule $a^0 = 1$ for any non-zero $a$:

$$2^0 = 1$$

The simplified expression is $\boxed{\mathbf{1}}$.

Using the rule $a^0 = 1$ for any non-zero $a$:

$$6 \times z^0 = 6 \times 1 = 6$$

The simplified expression is $\boxed{\mathbf{6}}$.

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$4^{-3} = \frac{1}{4^3}$$

Calculate the value:

$$\frac{1}{4^3} = \frac{1}{4 \times 4 \times 4} = \frac{1}{64}$$

The simplified expression is $\boxed{\mathbf{\frac{1}{64}}}$.

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$3x^{-3} = 3 \times \frac{1}{x^3} = \frac{3}{x^3}$$

The simplified expression is $\boxed{\mathbf{\frac{3}{x^3}}}$.

First, simplify the expression inside the parenthesis:

$$(1-4)^{-2} = (-3)^{-2}$$

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$(-3)^{-2} = \frac{1}{(-3)^2}$$

Calculate the value:

$$\frac{1}{(-3)^2} = \frac{1}{(-3) \times (-3)} = \frac{1}{9}$$

The simplified expression is $\boxed{\mathbf{\frac{1}{9}}}$.

First, simplify the expression inside the parenthesis:

$$(2-3)^{-1} = (-1)^{-1}$$

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$(-1)^{-1} = \frac{1}{(-1)^1} = \frac{1}{-1} = -1$$

The simplified expression is $\boxed{\mathbf{-1}}$.

Using the rule $a^m \div a^n = a^{m-n}$:

$$x^3 \div x^{-5} = x^{3 - (-5)} = x^{3+5} = x^8$$

The simplified expression is $\boxed{\mathbf{x^8}}$.

Using the rule $a^0 = 1$ for any non-zero $a$:

$$a^{-9} \div b^0 = a^{-9} \div 1 = a^{-9}$$

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$a^{-9} = \frac{1}{a^9}$$

The simplified expression is $\boxed{\mathbf{\frac{1}{a^9}}}$.

Using the rule $(ab)^n = a^n b^n$:

$$(3x)^{-3} = 3^{-3} x^{-3}$$

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$3^{-3} x^{-3} = \frac{1}{3^3} \times \frac{1}{x^3} = \frac{1}{27} \times \frac{1}{x^3} = \frac{1}{27x^3}$$

The simplified expression is $\boxed{\mathbf{\frac{1}{27x^3}}}$.

Multiply the coefficients and the variables separately:

$$9a^{-5} \times 4a^6 = (9 \times 4) \times (a^{-5} \times a^6)$$ $$= 36 \times a^{-5+6} = 36a^1 = 36a$$

The simplified expression is $\boxed{\mathbf{36a}}$.

Multiply the coefficients:

$$5 \times 4 \times 2 = 40$$

Multiply the variables using $a^m \times a^n = a^{m+n}$:

$$x^2 \times x^0 \times x^{-6} = x^{2+0+(-6)} = x^{2-6} = x^{-4}$$

Combine the results:

$$40x^{-4}$$

Using the rule $a^{-n} = \frac{1}{a^n}$:

$$40x^{-4} = \frac{40}{x^4}$$

The simplified expression is $\boxed{\mathbf{\frac{40}{x^4}}}$.

Rewrite the expression as a fraction:

$$\frac{15 \times 10^4}{3 \times 10^{-2}}$$

Separate the numerical parts and the powers of 10:

$$= \frac{15}{3} \times \frac{10^4}{10^{-2}}$$

Simplify the numerical part:

$$= 5 \times \frac{10^4}{10^{-2}}$$

Simplify the powers of 10 using $a^m \div a^n = a^{m-n}$:

$$= 5 \times 10^{4 - (-2)} = 5 \times 10^{4+2} = 5 \times 10^6$$

The simplified expression is $\boxed{\mathbf{5 \times 10^6}}$.