Properties of Exponents

An exponent is a number that indicates how many times to multiply the base by itself. For example, in $$x^3$$ , 3 is the exponent.

Cubing a number means raising it to the power of 3. For example, $$2^3 = 2 \times 2 \times 2 = 8$$ .

Squaring a number means raising it to the power of 2. For example, $$3^2 = 3 \times 3 = 9$$ .

When multiplying exponents with the same base, you add the exponents: $$x^a \times x^b = x^{a+b}$$ .

The power of a power rule states that when raising a power to another power, you multiply the exponents: $$\left(x^a\right)^b = x^{a \cdot b}$$ .

Using the power of a power rule: $$\left(3^2\right)^4 = 3^{2 \cdot 4} = 3^8$$ .

Using the product of powers rule: $$3^4 \times 3^5 = 3^{4+5} = 3^9$$ .

Using the power of a product rule: $$\left(2x^2\right)^3 = 2^3 \cdot (x^2)^3 = 8x^6$$ .

Any non-zero number raised to the power of zero equals one: $$x^0 = 1$$ (for $$x \neq 0$$ ).

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent: $$x^{-a} = \frac{1}{x^a}$$ .