In this lesson you’ll learn what exponents mean, the rules for positive/negative/zero exponents, and how to use scientific notation for large/small numbers.
An exponent shows repeated multiplication: 4 3 = 4 × 4 × 4 = 64 4^3 = 4 \times 4 \times 4 = 64 4 3 = 4 × 4 × 4 = 64
Rules:
a 1 = a a^1 = a a 1 = a a 0 = 1 a^0 = 1 a 0 = 1 a − n = 1 a n a^{-n} = \frac{1}{a^n} a − n = a n 1
Example:
2 − 3 = 1 2 3 = 1 8 2^{-3} = \frac{1}{2^3} = \frac{1}{8} 2 − 3 = 2 3 1 = 8 1 Multiplying powers: a m × a n = a m + n a^m \times a^n = a^{m+n} a m × a n = a m + n
Dividing: a m ÷ a n = a m − n a^m \div a^n = a^{m-n} a m ÷ a n = a m − n
Power of a power: ( a m ) n = a m × n (a^m)^n = a^{m \times n} ( a m ) n = a m × n
Scientific notation : Write numbers as a × 10 b a \times 10^b a × 1 0 b 1 ≤ ∣ a ∣ < 10 1 \leq |a| < 10 1 ≤ ∣ a ∣ < 10
4500 = 4.5 × 10 3 4500 = 4.5 \times 10^3 4500 = 4.5 × 1 0 3 0.00032 = 3.2 × 10 − 4 0.00032 = 3.2 \times 10^{-4} 0.00032 = 3.2 × 1 0 − 4
3 4 = 3 × 3 × 3 × 3 = 81 5 0 = 1 2 − 2 = 1 2 2 = 1 4 6 × 10 5 = 600,000 \begin{aligned}
3^4 &= 3 \times 3 \times 3 \times 3 = 81 \\
5^0 &= 1 \\
2^{-2} &= \frac{1}{2^2} = \frac{1}{4} \\
6 \times 10^5 &= 600{,}000
\end{aligned} 3 4 5 0 2 − 2 6 × 1 0 5 = 3 × 3 × 3 × 3 = 81 = 1 = 2 2 1 = 4 1 = 600 , 000 Simplify 4 3 × 4 2 4^3 \times 4^2 4 3 × 4 2
4 3 × 4 2 = 4 3 + 2 = 4 5 = 1024 4^3 \times 4^2 = 4^{3+2} = 4^5 = 1024 4 3 × 4 2 = 4 3 + 2 = 4 5 = 1024 Exponents appear in growth (population doubles = ×2 each period), interest (compound = P × (1 + r)^t), scientific measurements (3.2 × 10⁻⁴ grams), or computing (2¹⁰ bytes = 1 kilobyte). Understanding powers helps with large/small numbers and patterns.
Exponents are just “repeated multiplying.” Positive = big numbers, negative = fractions, zero = 1. The rules are shortcuts for multiplying/dividing powers. Practice with small exponents and real examples (growth, measurements) and it becomes intuitive fast.
What is $5^0$?
Calculate $3^{-2}$. A. $-9$ B. $\frac{1}{9}$ C. $9$ D. $-\frac{1}{9}$
Simplify $2^4 \times 2^3$. A. $2^7$ B. $2^{12}$ C. $2^1$ D. $4^7$
Write $0.00045$ in scientific notation. A. $4.5 \times 10^{-4}$ B. $4.5 \times 10^{-5}$ C. $4.5 \times 10^4$ D. $45 \times 10^{-5}$
Simplify $\frac{5^6}{5^4}$. A. $5^2 = 25$ B. $5^{10}$ C. $5^{24}$ D. $1$
Retry Quiz
