5-1 Powers and Exponents

Property

An exponent shows how many times the base is to be used as a factor. In the expression $5^4$, the base is 5 and the exponent is 4.

Examples

• $5^3 = 5 \cdot 5 \cdot 5 = 125$
• $10^4 = 10 \cdot 10 \cdot 10 \cdot 10 = 10,000$
• $(\frac{1}{3})^2 = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}$

Explanation

Think of an exponent as a tiny instruction manual for a number! The base is the number you're working with, and the exponent tells you exactly how many times to multiply that base by itself. It’s a super handy shortcut for writing out long, repetitive multiplications, saving you time and making your math look sleek and professional.

Property

Any repeated multiplication can be written in exponential form: $a \cdot a \cdot a \cdot \ldots \cdot a$ ($n$ factors) = $a^n$.

Conversely, any exponential expression can be expanded back into repeated multiplication. The base ($a$) indicates what number is being multiplied, and the exponent ($n$) indicates how many times the base appears as a factor.

Examples

• $5 \cdot 5 \cdot 5 \cdot 5 = 5^4$ (four factors of 5)
• $x^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x$ (six factors of $x$)
• $(-3) \cdot (-3) \cdot (-3) = (-3)^3$ (three factors of -3)

Property

The rule of repeated multiplication applies to any base, including decimals and fractions. For a fractional base, the exponent applies to both the numerator and the denominator.

Examples