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Learn Exponents

Repeated multiplication, plus the handful of rules that tame it.

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An exponent says how many times to multiply a number by itself: 2 to the fifth is 2 times 2 times 2 times 2 times 2, or 32. That simple idea grows into a set of rules for multiplying, dividing and raising powers, along with meanings for zero, negative and fractional exponents that keep the rules consistent. Exponents show up in area and volume, scientific notation, compound interest and every later algebra topic, so fluency with them saves effort everywhere. Learnr builds a focused exponents course and teaches each rule in short interactive lessons that show why it works, not just what it says.

How to learn Exponents, step by step

Write powers out in full at first. Expand x cubed times x squared as (x times x times x)(x times x) and count five factors. Seeing it written out shows why you add exponents when multiplying the same base.
Derive the quotient rule by cancelling. Write x to the fifth over x squared as five factors over two and cancel to get x cubed. Then notice that subtracting exponents gives the same answer every time.
Separate power of a power from products. (x cubed) squared means x cubed times x cubed, which is x to the sixth, so you multiply exponents. Contrast it with x cubed times x squared, which is x to the fifth.
Make sense of zero and negative exponents. Divide 2 to the third by 2 to the third to see why 2 to the zero must be 1. Continue the pattern 8, 4, 2, 1 downward to get 1/2 and 1/4 for the negative powers.
Link fractional exponents to roots. x to the 1/2 is the square root of x, and 8 to the 2/3 means take the cube root, 2, then square it, giving 4. Taking the root first keeps the numbers small.
Drill mixed problems with a daily habit. Problems that combine several rules are where errors hide. A short Learnr session every day, with a streak to keep it going, builds the reflex to apply rules in the right order.

What you'll learn

Evaluate powers, including negative bases with and without parentheses
Apply the product, quotient and power-of-a-power rules
Explain and use zero and negative exponents
Convert between fractional exponents and radicals
Write and compute with numbers in scientific notation

What's inside your Exponents course

A sample path Learnr builds for Exponents. Your real course adapts to your goal and level.

1. Powers, bases and exponents
2. Product and quotient rules
3. Power of a power, a product and a quotient
4. Zero and negative exponents
5. Fractional exponents and radicals
6. Scientific notation and exponential growth

How long does it take to learn Exponents?

The basic rules usually take 1 to 3 weeks at 2 to 4 hours a week. Fractional exponents and scientific notation add another week or two. Parents helping a child can cover whole-number powers in a few short sessions.

What you need before you start

Multiplication and division facts, working with fractions and some comfort with negative numbers. For the algebra rules, basic use of variables helps.

Common mistakes to avoid

Multiplying the base by the exponent, so 3 to the fourth becomes 12 instead of 81. An exponent counts repeated factors, not a single multiplication.
Treating a negative exponent as a negative number. 5 to the -2 is 1/25, a small positive number, not -25.
Adding exponents when bases differ, as in 2 cubed times 3 squared. The rules only combine powers that share the same base.

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Exponents questions, answered

Why is any number to the zero power equal to 1?

Because dividing a power by itself gives 1, and the quotient rule says the result is that base to the zero. Defining it as 1 keeps every exponent rule working. Zero to the zero is a special case usually left undefined in school.

How do I explain exponents to a child?

Use repeated multiplication with objects: 3 squared is 3 rows of 3 dots, and 2 cubed is a cube built from 8 blocks. Grow the exponent by one and notice the total multiplies each time.