Algebraic Exponents Practice

Practice using exponents in algebra. Exponents show repeated multiplication and appear often in algebra, functions, science, engineering, and calculus preparation.

Skill 1: Understanding Exponents

An exponent tells how many times a base is multiplied by itself. For example, x³ means x multiplied by itself three times: x × x × x.

Worked Example

Problem: Simplify x³ · x²

Step 1: Both terms have the same base, x.

Step 2: When multiplying like bases, add the exponents.

x³ · x² = x3 + 2

Step 3: Add.

x5

Answer: x5

Try These

  1. x² · x³
  2. a⁴ · a²
  3. y · y⁵
  4. m³ · m³
  5. b² · b · b⁴

Answer Key

  1. x5
  2. a6
  3. y6
  4. m6
  5. b7

Mastery Check

You are ready to move on when you can explain why multiplying like bases means adding the exponents.

Skill 2: Dividing Like Bases

When dividing expressions with the same base, subtract the exponents. The base stays the same. This rule works because division removes repeated factors from the numerator and denominator.

Worked Example

Problem: Simplify x⁵ ÷ x²

Step 1: Both terms have the same base, x.

Step 2: When dividing like bases, subtract the exponents.

x⁵ ÷ x² = x5 - 2

Step 3: Subtract.

x3

Answer: x3

Try These

  1. x⁶ ÷ x²
  2. a⁷ ÷ a³
  3. y⁵ ÷ y
  4. m⁸ ÷ m⁴
  5. b⁹ ÷ b²

Answer Key

  1. x4
  2. a4
  3. y4
  4. m4
  5. b7

Mastery Check

You are ready to move on when you can explain why dividing like bases means subtracting the exponents.

Skill 3: Power of a Power

When an exponent expression is raised to another power, multiply the exponents. This is called the power of a power rule.

Worked Example

Problem: Simplify (x²)³

Step 1: The expression x² is being raised to the 3rd power.

Step 2: Multiply the exponents.

(x²)³ = x2 × 3

Step 3: Multiply.

x6

Answer: x6

Try These

  1. (x³)²
  2. (a²)⁴
  3. (y⁵)²
  4. (m⁴)³
  5. (b⁶)²

Answer Key

  1. x6
  2. a8
  3. y10
  4. m12
  5. b12

Mastery Check

You are ready to move on when you can explain why a power raised to another power means multiplying the exponents.

Skill 4: Zero and Negative Exponents

A zero exponent means the value is 1, as long as the base is not zero. A negative exponent means the factor belongs in the denominator. Negative exponents do not make the value negative.

Worked Example

Problem: Simplify x-3

Step 1: A negative exponent means move the factor to the denominator.

x-3 = 1 / x3

Answer: 1 / x3

Try These

  1. x0
  2. a-2
  3. y-5
  4. m0
  5. b-1

Answer Key

  1. 1
  2. 1 / a2
  3. 1 / y5
  4. 1
  5. 1 / b

Mastery Check

You are ready to move on when you can explain that zero exponents become 1 and negative exponents move factors across the fraction bar.

Skill 5: Mixed Exponent Rules

Some problems use more than one exponent rule. Work one step at a time. First simplify powers, then multiply or divide like bases. Keep the base the same and carefully apply the correct exponent operation.

Worked Example

Problem: Simplify (x²)³ · x⁴

Step 1: Simplify the power of a power.

(x²)³ = x6

Step 2: Now multiply like bases by adding exponents.

x6 · x4 = x6 + 4

Step 3: Add.

x10

Answer: x10

Try These

  1. (x²)² · x³
  2. (a³)² ÷ a²
  3. y⁴ · y² ÷ y
  4. (m²)³ · m
  5. b⁸ ÷ b³ · b²

Answer Key

  1. x7
  2. a4
  3. y5
  4. m7
  5. b7

Mastery Check

You are ready to move on when you can identify which exponent rule is needed at each step instead of trying to solve the whole problem at once.

Back to Skills Home Previous: Linear Inequalities Next: Polynomials