Laws of Exponents

Exponents are a shorthand way of writing repeated multiplication. Instead of writing \(2 \times 2 \times 2 \times 2\), we write \(2^4\). The small raised number (4) is the exponent, and it tells us how many times to multiply the base (2) by itself.

When working with expressions that contain exponents, you need to follow specific rules. These aren't arbitrary—they come directly from what exponents mean.

Basic Exponent Notation

$$a^n = a \times a \times a \times ... \times a \text{ (n times)}$$

Example: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

Example: \(x^5 = x \times x \times x \times x \times x\)

The base can be any number or variable. The exponent must be a whole number for now (we'll talk about zero and negative exponents later).

Product Rule: Multiplying Powers with the Same Base

When you multiply two powers that have the same base, you add the exponents.

$$a^m \times a^n = a^{m+n}$$

Why? Because \(a^m\) means \(a\) multiplied by itself \(m\) times, and \(a^n\) means \(a\) multiplied by itself \(n\) times. When you multiply them together, you're multiplying \(a\) by itself \(m + n\) times total.

Example: \(x^3 \times x^4 = x^{3+4} = x^7\)

Think about it: \(x^3 = x \times x \times x\) and \(x^4 = x \times x \times x \times x\). Together that's \(x\) multiplied by itself 7 times.

Example: \(2^5 \times 2^3 = 2^{5+3} = 2^8 = 256\)

Example: \(y^2 \times y = y^2 \times y^1 = y^{2+1} = y^3\)

Notice that \(y\) by itself is the same as \(y^1\).

Quotient Rule: Dividing Powers with the Same Base

When you divide two powers with the same base, you subtract the exponents.

$$\frac{a^m}{a^n} = a^{m-n}$$

Example: \(\frac{x^7}{x^3} = x^{7-3} = x^4\)

This makes sense: \(\frac{x \times x \times x \times x \times x \times x \times x}{x \times x \times x}\). Three \(x\)'s cancel from top and bottom, leaving \(x^4\).

Example: \(\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625\)

Example: \(\frac{a^8}{a^5} = a^{8-5} = a^3\)

Power Rule: Raising a Power to a Power

When you raise a power to another power, you multiply the exponents.

$$(a^m)^n = a^{m \times n}$$

Example: \((x^2)^3 = x^{2 \times 3} = x^6\)

Why? \((x^2)^3\) means \(x^2 \times x^2 \times x^2\). Using the product rule, that's \(x^{2+2+2} = x^6\).

Example: \((3^2)^4 = 3^{2 \times 4} = 3^8 = 6561\)

Example: \((y^5)^2 = y^{10}\)

Power of a Product

When you raise a product to a power, you raise each factor to that power.

$$(ab)^n = a^n b^n$$

Example: \((2x)^3 = 2^3 x^3 = 8x^3\)

Example: \((3y)^4 = 3^4 y^4 = 81y^4\)

Example: \((xy)^5 = x^5 y^5\)

Power of a Quotient

When you raise a quotient to a power, you raise both numerator and denominator to that power.

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Example: \(\left(\frac{x}{2}\right)^3 = \frac{x^3}{2^3} = \frac{x^3}{8}\)

Example: \(\left(\frac{3}{y}\right)^2 = \frac{3^2}{y^2} = \frac{9}{y^2}\)

The Zero Exponent

Any non-zero number raised to the zero power equals 1.

$$a^0 = 1 \quad (a \neq 0)$$

This might seem strange, but it follows from the quotient rule. Consider \(\frac{a^3}{a^3}\). On one hand, anything divided by itself is 1. On the other hand, using the quotient rule: \(\frac{a^3}{a^3} = a^{3-3} = a^0\). So \(a^0 = 1\).

Example: \(5^0 = 1\)

Example: \(x^0 = 1\) (as long as \(x \neq 0\))

Example: \((3xy^2)^0 = 1\)

Negative Exponents

A negative exponent means "reciprocal."

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

Example: \(x^{-3} = \frac{1}{x^3}\)

Example: \(2^{-4} = \frac{1}{2^4} = \frac{1}{16}\)

Example: \(\frac{1}{y^{-2}} = y^2\)

To move a factor from numerator to denominator (or vice versa), change the sign of the exponent.

Combining Multiple Rules

Most problems require using several exponent rules together.

Example: Simplify \((x^2 y^3)^2\).

Apply power of a product: $$= x^4 y^6$$

Example: Simplify \(\frac{(2a^3)^2}{a^4}\).

Power of a product on top: $$= \frac{4a^6}{a^4}$$

Quotient rule: $$= 4a^{6-4} = 4a^2$$

Example: Simplify \(x^{-2} \times x^5\).

Product rule: $$= x^{-2+5} = x^3$$

Example: Simplify \(\frac{15x^7 y^2}{3x^3 y^5}\).

Divide coefficients: $$= 5 \times \frac{x^7}{x^3} \times \frac{y^2}{y^5}$$

Apply quotient rule: $$= 5x^4 y^{-3} = \frac{5x^4}{y^3}$$

Put It Into Practice

Simplify:

1. \(x^5 \times x^3\)
2. \(\frac{y^9}{y^4}\)
3. \((a^3)^4\)
4. \((2m)^3\)
5. \(n^0\)
6. \(x^{-5}\)
7. \(\frac{10a^8 b^3}{5a^2 b^7}\)
8. \((x^2 y^{-3})^2\)

Solutions:

1. \(x^8\)
2. \(y^5\)
3. \(a^{12}\)
4. \(8m^3\)
5. \(1\)
6. \(\frac{1}{x^5}\)
7. \(2a^6 b^{-4} = \frac{2a^6}{b^4}\)
8. \(x^4 y^{-6} = \frac{x^4}{y^6}\)

Where People Get Stuck

Adding exponents when multiplying different bases is impossible. You cannot simplify \(x^2 \times y^3\) using exponent rules because \(x\) and \(y\) are different bases. The expression stays as \(x^2 y^3\).

Multiplying exponents instead of adding when using the product rule is a surprisingly common error. When you see \(x^2 \times x^3\), you add the exponents to get \(x^5\), not multiply them to get \(x^6\).

Forgetting to apply the exponent to coefficients trips people up. When you calculate \((3x)^2\), both the 3 and the \(x\) get squared: \(9x^2\), not \(3x^2\).

Thinking \(a^0 = 0\) is wrong. Any non-zero number raised to the zero power equals 1, not 0. So \(5^0 = 1\), \(x^0 = 1\), and \((374)^0 = 1\).

Getting confused by negative exponents leads to mistakes. A negative exponent means "reciprocal," not "negative answer." So \(x^{-3} = \frac{1}{x^3}\), which is positive if \(x\) is positive.