Exponent Rules and Integer Exponents

Grade 8 · Math · Study Guide

Master the laws of exponents to simplify expressions with positive, negative, and zero exponents.

Exponents provide a compact way to write repeated multiplication. The expression x^n means x multiplied by itself n times. A handful of rules let you simplify complex expressions quickly.

Product Rule: x^m times x^n = x^(m+n). Quotient Rule: x^m divided by x^n = x^(m-n). Power Rule: (x^m)^n = x^(m times n). Zero exponent: x^0 = 1 for any nonzero x. Negative exponent: x^(-n) = 1/x^n — take the reciprocal.

These rules work together. Always apply one rule at a time and watch signs carefully. These laws are essential for scientific notation, polynomials, and beyond.

Laws of Exponents Reference
Rule	Formula	Example
Product Rule	x^m * x^n = x^(m+n)	x^3 * x^4 = x^7
Quotient Rule	x^m / x^n = x^(m-n)	x^6 / x^2 = x^4
Power Rule	(x^m)^n = x^(mn)	(x^2)^3 = x^6
Zero Exponent	x^0 = 1	5^0 = 1
Negative Exponent	x^(-n) = 1/x^n	2^(-3) = 1/8
Product to Power	(xy)^n = x^n * y^n	(2x)^3 = 8x^3

Key terms

Base: The number or variable being raised to a power; the x in x^n.
Exponent (Power): How many times to multiply the base by itself.
Product Rule: x^m * x^n = x^(m+n) — add exponents, same base.
Quotient Rule: x^m / x^n = x^(m-n) — subtract exponents, same base.
Power Rule: (x^m)^n = x^(m*n) — multiply exponents.
Zero Exponent: x^0 = 1 for any x not equal to 0.
Negative Exponent: x^(-n) = 1/x^n — move to denominator, make exponent positive.

Flashcards

Product rule: x^m * x^n = x^(m+n). Same base — add exponents.
Quotient rule: x^m / x^n = x^(m-n). Same base — subtract exponents.
Power rule: (x^m)^n = x^(m*n). Raise a power to a power — multiply exponents.
Zero exponent rule: x^0 = 1 for any x not equal to 0.
Negative exponent rule: x^(-n) = 1/x^n. Move to denominator, make positive.
Power of a product: (xy)^n = x^n * y^n. Distribute exponent to each factor.

Practice questions

Simplify: x^5 * x^3
1. x^15
2. x^8
3. x^2
4. 2x^8
Show answer
Answer: x^8
Product rule: x^5 * x^3 = x^(5+3) = x^8.
Any nonzero number raised to the power of 0 equals 0.
Show answer
Answer: False
x^0 = 1 (not 0) for any nonzero x.
2^(-4) = 1/___.
Show answer
Answer: 16
2^(-4) = 1/2^4 = 1/16.
Simplify: (x^3)^4
1. x^7
2. x^12
3. x^81
4. 4x^3
Show answer
Answer: x^12
Power rule: (x^3)^4 = x^(3*4) = x^12.
Simplify: y^8 / y^3
1. y^5
2. y^11
3. y^24
4. y^2
Show answer
Answer: y^5
Quotient rule: y^8 / y^3 = y^(8-3) = y^5.

Fun facts

Any nonzero number to the power of 0 equals 1 — even (-47)^0 = 1. This follows from the quotient rule: x^n / x^n = x^0 = 1.
Computer storage uses powers of 2: 2^10 = 1,024 (about 1 kilobyte) and 2^20 is about 1 megabyte.

Get a personalized course on Exponent Rules and Integer Exponents built around your child.