Basic Exponent Manipulations

Summary:

The following formulas will allow you to manipulate exponents to solve many different problems: $$x^a*x^b =x^{a+b}$$ $$\frac{x^a}{x^b} = x^{a-b}$$ $$x^{a^b} = x^{a*b}$$ $$a^{-b} = \frac{1}{a^{b}}$$

The manipulation of exponents is important, as it will eventually allow you to solve for variables in exponential equations. It is important to follow order of operations while soving these problems.

Example 1:

Evaluate the following:

\(1. (2x^4)(3x^3) \)
\(2. \frac{6x^8}{2x^4}\)
\(3. (2x)^{2^3}\)

Part 1:

We can first rewrite this expression as: $$2*3*x^4*x^3$$ Which is equal to: $$6*x^4*x^3$$ Finally, we can use the formula to get: $$6x^{4+3} = 6x^7$$

Part 2:

We can first rewrite this as: $$\frac{6}{2}*\frac{x^8}{x^4}$$ Which is equal to: $$3*\frac{x^8}{x^4}$$ Next, we can use the second formula to get: $$3x^{8-4} = 3x^4$$

Part 3:

We can use the third formula to get: $$(2x)^{2*3} = (2x^6)$$ This is equal to: $$2^6*x^6 = 64x^6$$

These formulas also work when multiple variables are involved, as you will see in the next example

Example 2:

\(1. (3x^2y^3)(7x^5y^2) \)
\(2. \frac{12x^3y^2}{4x^2y^4}\)
\(3. (xy^2)^{2^3}\)

Part 1:

We can first rewrite this expression as: $$3*7*x^2*x^5*y^3*y^2$$ Which is equal to: $$21*x^2*x^5*y^3*y^2$$ Finally, we can use the formula to get: $$21x^{2+5}y^{3+2} = 21x^7y^5$$

Part 2:

We can first rewrite this as: $$\frac{12}{4}*\frac{x^3}{x^2}*\frac{y^2}{y^4}$$ Which is equal to: $$3*\frac{x^3}{x^2}*\frac{y^2}{y^4}$$ Next, we can use the second formula to get: $$3x^{3-2}y^{2-4} = 3xy^{-2}$$

Part 3:

We can use the third formula to get: $$(xy^2)^{2*3} = ((xy^2)^6)$$ This is equal to: $$x^6*(y^2)^6$$ Now, we have to apply the formula again to the y term, to get: $$x^6(y^{2*6}) = x^6y^{12}$$

The fourth rule is about negative exponents, and will be covered in the next example.

Example 3:

Evaluate the following:

\(1. 2^{-2}\)
\(2. \frac{x^{-4}}{x^2}\)
\(3. \frac{x^2}{x^{-3}}\)

Part 1:

First, we can rewrite this using the fourth formula as: $$\frac{1}{2^2}$$ Which is equal too: $$\frac{1}{4}$$
Part 2:

Using the fourth formula, we can rewrite this as: $$x^{-4}*x^{-2}$$ Then, we can use the first formula to get: $$x^{-4+(-2)} = x^{-6}$$
Part 3:

Using the fourth formula, we can rewrite this as: $$x^{2}*x^{-(-3)} = x^2*x^3$$ Then, we can use the first formula to get: $$x^{2+3} = x^{5}$$

Conlcusion:
In conclusion, you have learned four rules allowing you to simplify exponential expressions. These rules will later prove useful when you need to solve exponential equations, or perform calculations using exponents

Next: Multiplying polynomials