Exponents: Step-by-Step Guide
An exponent tells you how many times to multiply a number by itself. In 2^3, the 2 is the base and the 3 is the exponent, so 2^3 = 2 x 2 x 2 = 8. A few simple rules let you combine and simplify powers without writing out every multiplication.
Math Racer skill guide · Middle School · Grades 6-8
How to solve it, step by step
- Find the base and the exponent. The base is the number being multiplied. The exponent is how many times it is multiplied. Look at the problem and decide which rule you need.\[a^{n} = \underbrace{a \cdot a \cdots a}_{n \text{ times}}\]
- Multiplying or dividing with the same base. When you multiply powers that have the same base, add the exponents. When you divide powers with the same base, subtract the exponents.\[a^{m} \cdot a^{n} = a^{m+n},\quad \frac{a^{m}}{a^{n}} = a^{m-n}\]
- Power of a power, zero, and negative exponents. A power raised to another power: multiply the exponents. Any number (except 0) to the power of 0 is 1. A negative exponent means 'one over' the positive power.\[(a^{m})^{n} = a^{mn},\quad a^{0} = 1,\quad a^{-n} = \frac{1}{a^{n}}\]
- Simplify and work out the answer. Use the rules to combine everything into one exponent. If the exponent is small, work out the final number.\[a^{m} \cdot b^{m} = (ab)^{m}\]
Worked example
Simplify: (2^3 \cdot 2^4) / 2^5
Step 1: Every power has the same base, which is 2.
Step 2: Top of the fraction: multiplying with the same base, so add the exponents. 2^3 x 2^4 = 2^(3+4) = 2^7.
Step 3: Now divide with the same base, so subtract the exponents. 2^7 / 2^5 = 2^(7-5) = 2^2.
Step 4: Work out the power: 2^2 = 2 x 2 = 4.
Step 5: The answer is 4.
Check: Do it the long way. 2^3 = 8, 2^4 = 16, and 2^5 = 32. Then (8 x 16) / 32 = 128 / 32 = 4.
Tips and common mistakes
- The add-the-exponents rule only works when the bases are the same. 2^3 x 3^2 cannot be combined this way.
- Any number except 0 raised to the power of 0 equals 1.
- A negative exponent does not make the answer negative. It flips the number: 2^(-3) = 1/2^3 = 1/8.
- Watch the parentheses. (-3)^2 = 9, but -3^2 = -9, because the exponent only applies to the 3.
- A product to a power: (ab)^n = a^n x b^n. This does not work for sums. (a + b)^2 is not a^2 + b^2.
- If you are not sure about a rule, test it with small numbers like 2^2 x 2^3 = 4 x 8 = 32 = 2^5.