pAIrmatic Lessons
UNIT 1Exponents and rootsLesson 1 of 31 Exponent Rules: Product, Quotient and Power2 Zero and Negative Exponents3 Square Roots and Cube Roots

Exponent Rules: Product, Quotient and Power

8.EE.1Grade 8Expressions & Equations20 practice questions

An exponent is a short way to write the same number multiplied again and again. A few rules help you work with exponents quickly.

What you'll be able to do

The big idea

2⁵ is a short way to write 2 × 2 × 2 × 2 × 2. The 2 is called the base. The small raised 5 is the exponent. It tells you how many 2s to multiply.

Multiplying. Look at 2³ × 2⁴. That is three 2s times four more 2s. Count them all: 3 + 4 = 7. So 2³ × 2⁴ = 2⁷. When the bases match, you add the exponents.

Dividing. Look at 2⁶ ÷ 2². Write six 2s on top and two 2s on the bottom. Each 2 on the bottom cancels a 2 on top. That leaves 6 − 2 = 4 of them. So 2⁶ ÷ 2² = 2⁴. When the bases match, you subtract the exponents.

Power of a power. Look at (2²)³. It means 2² × 2² × 2². Add the exponents: 2 + 2 + 2 = 6. That is the same as 2 × 3. So (2²)³ = 2⁶. You multiply the exponents.

More than one letter. Work on the numbers first. Then work on each letter on its own. For example, (3x⁴y)(5x²y³) = 15x⁶y⁴. A power outside parentheses goes to every part inside. So (2x³)² = 4x⁶, because the 2 gets squared too.

Watch the signs. An exponent covers only what it touches. So −3² = −9, but (−3)² = 9.

Changing the base. The rules need matching bases. 4 is 2², so 4³ = (2²)³ = 2⁶. Now 4³ × 2⁵ = 2⁶ × 2⁵ = 2¹¹.

Finding a missing exponent. If 2^n = 2⁷, then n = 7. Make the bases match, then compare the exponents.

Adding powers is different. The rules work for multiplying and dividing, not adding. 2⁵ + 2⁵ is two copies of 2⁵. That is 2 × 2⁵ = 2⁶, not 2¹⁰.

Words you need

Base
The number or letter being multiplied. In 5³, the base is 5.
Exponent
The small raised number. In 5³, the exponent is 3. It says to multiply three 5s.
Power
A base with an exponent, like 5³. You say it as "5 to the third power."
Coefficient
The number in front of a letter. In 4x², the coefficient is 4.
Simplify
Write something in its shortest form. x³ · x⁴ simplifies to x⁷.
Product and quotient
A product is the answer to a times problem. A quotient is the answer to a divide problem.
Square root and cube root
The square root of 9 is 3, because 3 × 3 = 9. The cube root of 8 is 2, because 2 × 2 × 2 = 8.

The rules

When you...Do thisExample
Multiply powers with the same baseAdd the exponentsx³ · x⁴ = x⁷
Divide powers with the same baseSubtract the exponentsx⁶ ÷ x² = x⁴
Raise a power to a powerMultiply the exponents(x²)³ = x⁶
Raise a product to a powerRaise every part(2x³)² = 4x⁶
Have different basesRewrite them to match4³ = 2⁶

Worked examples

Example 1

Simplify (2x³y)(4x²y⁵).

  1. Multiply the numbers: 2 × 4 = 8.
  2. Add the exponents on x: 3 + 2 = 5.
  3. y by itself means y¹. Add the exponents on y: 1 + 5 = 6.
  4. So the answer is 8x⁵y⁶.

Answer: 8x⁵y⁶

Example 2

Simplify (3a²b)³.

  1. The power 3 goes to every part inside the parentheses.
  2. The 3 gets cubed: 3 × 3 × 3 = 27.
  3. For a², multiply the exponents: 2 × 3 = 6. For b, it is 1 × 3 = 3.
  4. So the answer is 27a⁶b³.

Answer: 27a⁶b³

Example 3

Write 8² × 2³ as one power of 2.

  1. 8 is 2³, so 8² = (2³)².
  2. Multiply the exponents: 3 × 2 = 6. So 8² = 2⁶.
  3. Now the bases match: 2⁶ × 2³ = 2⁹.

Answer: 2⁹ = 512

Common mistakes

Multiplying the exponents when you multiply powers: x³ · x⁴ = x¹². Add them instead. x³ · x⁴ = x⁷. You multiply exponents only for a power of a power.

Multiplying the bases: 2³ × 2⁴ = 4⁷. The base stays the same. You are still multiplying 2s. The answer is 2⁷.

Forgetting the number inside the parentheses: (2x³)² = 2x⁶. The 2 gets squared too. 2² = 4, so (2x³)² = 4x⁶.

Reading −3² as 9. The exponent covers only the 3. So −3² = −9. Only (−3)² = 9.

Practice: 20 questions

5 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

  1. Simplify (3x⁴y)(5x²y³).
  2. Simplify (2a³b²)⁴.
  3. Simplify 20m⁹n⁴ ÷ (5m³n).
  4. A kilometer is 10³ meters. A meter is 10² centimeters. A centimeter is 10 millimeters. How many millimeters are in 10⁴ kilometers? Write your answer as a power of 10. Word problem
  5. Sam says 2³ × 2⁴ = 4⁷. Find his mistake. What is the correct answer?

Tier 2 Intermediate One more step to think through

  1. Simplify (4x³y²)³ ÷ (8x⁵y).
  2. Simplify −3x² · (−3x)².
  3. Which is greater, −2⁴ or (−2)⁴?
  4. A square tile is 3a²b units long on each side. What is the total area of 12 of these tiles? Word problem
  5. A box holds 4³ bags. Each bag holds 2⁵ marbles. How many marbles are in the box? Write your answer as one power of 2. Word problem

Tier 3 Advanced Several steps - find the hidden structure

  1. Simplify (2a³b)⁴ · (3ab²)² ÷ (6a⁵b³).
  2. (3xᵏy²)³ ÷ (9x⁴y) = 3x¹¹y⁵. What is k?
  3. A colony starts with 81 bacteria. The number triples every hour. After how many hours will there be 3¹⁰ bacteria? Word problem
  4. A data center has 2⁴ server cabinets. Each cabinet holds 4³ hard drives, and each drive stores 8² gigabytes. What is the total storage of the data center, written as a single power of 2? Word problem
  5. Jun says 3⁴ · 9² = 27⁶. Find his mistake. What is the correct answer, as a power of 3?

Tier 4 Challenge Stretch problems, some above grade level

  1. Put 2³⁰⁰, 3²⁰⁰ and 5¹⁰⁰ in order from least to greatest. Do it without working them out.
  2. How many times as great as (3²)³ is 3 raised to the power 2³?
  3. Find the value of (2¹⁰⁰ − 2⁹⁹) ÷ 2⁹⁸.
  4. Solve 3ˣ⁺² + 3ˣ = 90.
  5. Raising a number to the power 1/3 takes its cube root. For example, 8^(1/3) = 2, because 2 × 2 × 2 = 8. Simplify (27x⁶y⁹)^(2/3). Above grade level

Answers

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