Algebra 1 Exponent Rules Infographic

First, What is an Exponent?

An exponent tells you how many times to multiply a number (the base) by itself.

4 ³

→

4 is the Base

(The number being multiplied)

3 is the Exponent

(How many times to multiply the base)

Let's solve: 4³

Step 1: Identify the parts.

The Base is 4. The Exponent is 3.

Step 2: Understand the instruction.

The exponent '3' tells the base '4' to multiply itself 3 times.

Step 3: Write it out and solve.

4 × 4 × 4 = 16 × 4 = 64

Result:

4³ = 64

The Three Core Rules

1. Product Rule

When multiplying two terms with the same base, you ADD their exponents.

a^m · aⁿ = a^m+n

Easy (No Coeff.): (x³)(x⁴)

Step 1: Keep the base the same.

Base is x

Step 2: Add the exponents.

x³⁺⁴ = x⁷

Result:

x⁷

Medium (With Coeff.): (3x²)(4x⁵)

Step 1: Multiply the coefficients.

3 · 4 = 12

Step 2: Add the exponents of the same base.

x²⁺⁵ = x⁷

Result: Combine the parts.

12x⁷

Harder (Multi-variable): (-2a³bc²)(5ab⁴c)

Step 1: Multiply the coefficients.

-2 · 5 = -10

Step 2: Add exponents for each base.

a³⁺¹=a⁴ | b¹⁺⁴=b⁵ | c²⁺¹=c³

Result: Combine everything.

-10a⁴b⁵c³

2. Quotient Rule

When dividing two terms with the same base, you SUBTRACT the exponents.

a^m aⁿ

= a^m-n

Easy (No Coeff.):

y⁷y³

Step 1: Keep the base the same.

Base is y

Step 2: Subtract exponents (top minus bottom).

y^7-3 = y⁴

Result:

y⁴

Medium (With Coeff.):

10y⁸2y⁵

Step 1: Divide the coefficients.

10 ÷ 2 = 5

Step 2: Subtract the exponents.

y^8-5 = y³

Result: Combine the parts.

5y³

Harder (Multi-variable):

15x⁷y³z²3x²yz

Step 1: Divide the coefficients.

15 ÷ 3 = 5

Step 2: Subtract exponents for each base.

x^7-2=x⁵ | y^3-1=y² | z^2-1=z¹

Result: Combine everything.

5x⁵y²z

3. Power Rule

When raising a power to another power, you MULTIPLY the exponents.

(a^m)ⁿ = a^m·n

Easy (No Coeff.): (b⁵)²

Step 1: Keep the base the same.

Base is b

Step 2: Multiply the exponents.

b^5·2 = b¹⁰

Result:

b¹⁰

Medium (With Coeff.): (2b⁴)³

Step 1: Apply the outer exponent to the coefficient.

2³ = 8

Step 2: Multiply the variable's exponent by the outer exponent.

b^4·3 = b¹²

Result: Combine the parts.

8b¹²

Harder (Multi-variable): (2m³n²p)⁴

Step 1: Apply the outer exponent to the coefficient.

2⁴ = 16

Step 2: Multiply exponents for each base.

m^3·4=m¹² | n^2·4=n⁸ | p^1·4=p⁴

Result: Combine everything.

16m¹²n⁸p⁴