# Properties of exponents

## Big Picture

When an exponent is a positive whole number, an exponent is a shorthand way to represent large quantities of multiplication. Exponents can represent any value that is multiplied by itself repeatedly (for example, an expression such as $$5·5·5·5$$). Variables can have exponents just like real numbers. When the exponent is zero, negative, or fractional, there are certain rules we have to remember.

## Key Terms

Power: An expression with a base and an exponent.

Exponent: A number or symbol written as the superscript to the upper right of another number that is a type of mathematical operation.

## Exponential Form

For the power $$x^n$$

• $$x$$ is the base
• $$n$$ is the exponent (or the power)

We don’t usually write out the exponent if $$n = 1$$, so $$x^1 = x$$. If $$n = 2$$, we say $$x$$ squared. If $$n = 3$$, we say $$x$$ cubed.

If $$n$$ is a positive whole number, an exponent is a short-hand notation for repeated multiplication.

• Example: $$x^5 = x ∙ x ∙ x ∙ x ∙ x$$
• Example: $$(3a)^4 = (3a)(3a)(3a)(3a)$$, which can be simplified: $$3 · 3 · 3 · 3 · a · a · a · a = 81a^4$$

Exponents of negative numbers:

• Even powers of negative numbers are positive.
$$(-2)^6 = (-2)(-2)(-2)(-2)(-2)(-2) = \underbrace{(-2)(-2)}_{+4} . \underbrace{(-2)(-2)}_{+4} . \underbrace{(-2)(-2)}_{+4} = +64$$
• Odd powers of negative numbers are negative.
$$(-2)^5 = (-2)(-2)(-2)(-2)(-2) = \underbrace{(-2)(-2)}_{+4} . \underbrace{(-2)(-2)}_{+4} . \underbrace{(-2)}_{-2} = -32$$

## Properties Involving Products

### Product Rule for Exponents

To multiply powers with the same base, add the exponents together.

• $$x^m · x^n = x^{m+n}$$

For example, $$x^5 · x^3$$ can be written out as:

$$\underbrace{x . x . x . x . x}_{x^5} . \underbrace{x . x . x}_{x^3} = \underbrace{x . x . x . x . x . x . x . x}_{x^8}$$ Don’t multiply the bases! $$22 ∙ 23 = 25$$, NOT $$45$$.

### Power Rule for Exponents

To take a power of a product, multiply the exponents together.

• $$(x^m)^n · x^{mn}$$
• $$(xy)^m · x^my^m$$

## Properties Involving Quotients

### Quotient Rule for Exponents

To divide powers with the same base, subtract the exponents.

• $$\frac{x^m}{x^n} = x^{m-n}$$

### Power Rule for Quotients

To take a power of a quotient, multiply that exponent to the exponent of the numerator and the exponent of the denominator.

• $$(\frac{x^m}{x^n})^p = \frac{x^{mp}}{x^{np}}$$

# Properties of exponents Cont.

## Zero, Negative, Fractional Exponents

### Zero Rule for Exponents

Any number raised to the zero power is equal to 1.

• $$x^0 = 1$$

If you can’t remember this, use the quotient rule where the exponents of the numerator and the denominator are equal.

$$\frac{x^n}{x^n} = x^{n-n} = x^0 = 1$$

### Negative Power Rule for Exponents

A number raised to a negative number is the same as a fraction with the exponent in the denominator.

• $$x^{-n} = \frac{1}{x^n} \text{ and } x \neq 0$$

Example: Consider $$2^{-2} = \frac{1}{2^2}$$

### Negative Power Rule for Fractions

A fraction raised to a negative number is the same as taking the reciprocal of the fraction, then raising the numerator and denominator to the same exponent.

• $$(\frac{x}{y})^{-n} = (\frac{y}{x})^n = (\frac{y^n}{x^n}), \text{ where } x \neq 0 \text{ and } y \neq 0$$

### Rule for Fractional Exponents

A number raised to a fractional number is the same as taking the root of it.

• $$x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$$