Packets  /  Exponent Rules
Grades 7-10Algebra

Exponent Rules

Simplify anything with exponents — the product, quotient, and power rules, zero and negative exponents, fractional exponents, and scientific notation. Learn the rules, practice with instant feedback, and beat the 60-second challenge.

1The Product Rule

Key idea: When you multiply powers with the SAME base, keep the base and ADD the exponents.
  1. Check the bases are identical (x and x, or 2 and 2).
  2. Keep the base exactly as it is — do not multiply the bases.
  3. Add the exponents.
  4. Simplify the result if asked for a number.
Worked example
x3 · x4
Same base x, so ADD the exponents: x3+4 = x7. (Count the x's: three x's times four x's is seven x's multiplied.)
Worked example
25 · 23
Same base 2: 25+3 = 28. As a number: 28 = 256. Keep it as 28 or evaluate — both are correct.
Watch out: This rule only works for the SAME base! 2 cubed times 3 squared is NOT 6 to the 5th — compute it directly: 8 times 9 = 72. Also, multiplication ADDS exponents; it does not multiply them (that is the power rule).

2The Quotient Rule

Key idea: When you divide powers with the SAME base, keep the base and SUBTRACT the exponents — top minus bottom.
  1. Check the bases are identical.
  2. Keep the base.
  3. Subtract the bottom exponent from the top exponent (top minus bottom!).
  4. Simplify if needed.
Worked example
37 / 34
Same base 3, division: SUBTRACT: 37-4 = 33 = 27.
Worked example
x9 / x2
x9-2 = x7.
Watch out: Order matters: it is top minus bottom, never bottom minus top. And note what happens with x to the 5th over x to the 5th: the exponent becomes 0 — which leads to the next rule.

3The Power Rule

Key idea: A power raised to another power: keep the base and MULTIPLY the exponents.
  1. Spot the pattern: a power inside parentheses, raised to another power outside.
  2. Keep the base.
  3. Multiply the two exponents.
  4. Evaluate if it becomes a plain number.
Worked example
(y2)5
Power of a power: MULTIPLY the exponents: y2·5 = y10.
Worked example
(23)2
23·2 = 26 = 64.
Watch out: A power of a power MULTIPLIES exponents — it does not add them. If the inside is a sum, like (x squared plus x cubed) squared, the rule does NOT apply — the base must be a single power.

4Powers of Products and Quotients

Key idea: An exponent outside parentheses hits EVERY factor inside — raise each factor separately, then combine.
  1. For a product: raise each factor separately, then multiply.
  2. For a quotient: raise the numerator and the denominator separately.
  3. Apply the power rule (xm)n = xmn to any power factors.
  4. Combine and simplify.
Worked example
(3x2)3
Give EVERY factor inside the exponent: 33 · (x2)3 = 27 · x6 = 27x6.
Worked example
252
Square the top and the bottom: 22 / 52 = 4/25.
Watch out: The #1 mistake: (3x) squared = 9x squared, NOT 3x squared — the 3 gets squared too! Every factor inside the parentheses receives the outside exponent, including plain numbers.

5The Zero Exponent

Key idea: Anything nonzero to the 0 power equals 1. Why? x to the 5th over x to the 5th is x to the 0 by the quotient rule, but anything over itself is 1 — so x to the 0 must be 1.
  1. Check the base is not zero.
  2. If the base is nonzero, the answer is simply 1.
  3. For a product base like (2x)0, the 1 covers the whole thing.
Worked example
70
Any nonzero base to the 0 power is 1: 70 = 1.
Worked example
(2x)0
The whole base (2x) is nonzero, so (2x)0 = 1.
Watch out: 0 to the 0th is undefined, not 1. And watch the signs: (negative 5) to the 0th is 1, but negative 5 to the 0th means negative (5 to the 0th), which is negative 1. Parentheses decide what the exponent applies to!

6Negative Exponents

Key idea: A negative exponent does NOT make the answer negative — it means move the factor to the other side of the fraction bar and make the exponent positive.
  1. Find the factor with the negative exponent.
  2. Move it across the fraction bar (numerator to denominator, or denominator to numerator).
  3. Flip the sign of its exponent.
  4. Simplify the remaining positive exponents.
Worked example
4-2
Negative exponent means FLIP: 4-2 = 1 / 42 = 1/16.
Worked example
1 / x-3
Moving x-3 up flips the sign: 1 / x-3 = x3.
Watch out: 4 to the negative 2nd equals 1/16, which is POSITIVE. The minus sign means flip, never a negative answer. Also: only the factor with the negative exponent moves — in 3 times x to the negative 2nd, only the x flips: 3 over x squared.

7Fractional Exponents

Key idea: Fractional exponents are roots in disguise: the DENOMINATOR of the fraction is the root to take, the numerator is the power to raise to.
  1. Read the denominator of the exponent — that is the root to take.
  2. Read the numerator — that is the power to raise to.
  3. Easiest path: take the root first (small numbers!), then raise to the power.
  4. Check: does the root exist as a whole number? That is a hint you chose right.
Worked example
81/3
Denominator 3 means cube root: 81/3 = cube root of 8 = 2.
Worked example
43/2
Denominator 2 means square root: 43/2 = (square root of 4)3 = 23 = 8.
Watch out: The denominator is the ROOT, not a division. 8 to the 1/3 equals 2, not 8 divided by 3. Memorize: bottom of the fraction = which root.

8Scientific Notation

Key idea: Scientific notation writes big or tiny numbers as a times 10 to the n, where a is between 1 and 10. Positive n for big numbers, negative n for small ones.
  1. Move the decimal point until exactly one nonzero digit sits left of it — that is a.
  2. Count how many places you moved the decimal — that is n.
  3. Moved left (big number): n is positive. Moved right (small number): n is negative.
  4. To convert back: positive n moves the decimal right, negative n moves it left.
Worked example
4,500 in scientific notation
Move the decimal 3 places left: 4.5 × 103. Check: 4.5 × 1000 = 4,500.
Worked example
0.0003 in scientific notation
Move the decimal 4 places right: 3 × 10-4. Check: 3 / 10,000 = 0.0003.
Watch out: The coefficient a must be between 1 and 10 (not including 10). 45 times 10 squared is NOT scientific notation — 4.5 times 10 cubed is. And a negative exponent here means a small number, not a negative number.
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60-Second Challenge

How many exponent rules problems can you solve in 60 seconds?