Integer Exponents

Grade 8 · algebra · 100 practice problems · read aloud

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Integer Exponents: The Power of Numbers 🚀

An exponent tells you how many times to multiply a number (the base) by itself. For example, in 5³, 5 is the base and 3 is the exponent: 5 × 5 × 5 = 125. This is a shortcut for repeated multiplication, making it easier to write and calculate with very large or very small numbers.

Step-by-Step Guide

  1. Identify the base and exponent. What number is being multiplied?
  2. Multiply the base by itself as many times as the exponent says.
  3. Special Case: Negative Exponents mean "take the reciprocal." A negative exponent does NOT make the answer negative! Instead, move the base to the denominator: x⁻ⁿ = 1/xⁿ.
  4. Special Case: Zero Exponent Any non-zero number to the power of zero equals 1. (e.g., 8⁰ = 1)

Visual Examples

Example 1: Positive Exponent

Simplify: 4²

Step 1: Base is 4, exponent is 2.

Step 2: 4 × 4 = 16

Example 2: Negative Exponent

Simplify: 3⁻³

Step 1: Base is 3, exponent is -3.

Step 2: Take the reciprocal to make the exponent positive: 1 / 3³

Step 3: Simplify 3³: 3 × 3 × 3 = 27.

Answer: 1/27

Example 3: With a Negative Base

Simplify: (-2)⁴

Step 1: The base is -2 (in parentheses!), exponent is 4.

Step 2: (-2) × (-2) × (-2) × (-2) = 4 × 4 = 16

Note: An even exponent makes a negative base positive.

Common Mistakes to Avoid ⚠️

Confusing Negative Bases and Negative Exponents: (-5)² = 25, but -5² = -25. The exponent only applies to what is directly before it. Parentheses matter!

Misunderstanding Negative Exponents: 2⁻³ is NOT -8. It is 1 / 2³ = 1/8. A negative exponent means "reciprocal," not "negative answer."

Forgetting the Zero Rule: Any non-zero number to the zero power is 1. 10⁰ = 1, (-5)⁰ = 1.

Tips & Tricks

  • Reciprocal Flip: Remember the phrase "Flip the base, change the sign" for negative exponents.
  • Even/Odd Check: A negative base with an even exponent gives a positive answer. A negative base with an odd exponent gives a negative answer.
  • Zero Power Shortcut: If you see a number to the zero power, the answer is always 1 (as long as the base isn't zero).

How to Practice

Master integer exponents with these activities:

  • Flashcards: Create cards with different bases and exponents (positive, negative, and zero).
  • Practice Worksheets: Start with simple positive exponents, then mix in negatives and zeros.
  • Real-World Connection: Look for exponents in the wild! They are used in science for things like scientific notation (e.g., the distance to the sun is about 1.5 × 10⁸ km).

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 (3² × 3⁴) ÷ 3³ = ?

Hint: When multiplying terms with the same base, add the exponents. When dividing, subtract the exponents. Remember to apply the order of operations correctly.

Show the answer

Answer: 27

  1. Simplify inside the parentheses first. We have 3² × 3⁴. When multiplying powers with the same base, we add the exponents: 3² × 3⁴ = 3^(2 + 4) = 3⁶.
  2. Now the expression is 3⁶ ÷ 3³. When dividing powers with the same base, we subtract the exponents: 3⁶ ÷ 3³ = 3^(6 − 3) = 3³.
  3. Calculate 3³. 3³ = 3 × 3 × 3 = 9 × 3 = 27. Final Answer: 27

2 (3² × 3³) ÷ 3⁴ = ?

Hint: When multiplying terms with the same base, add the exponents. When dividing, subtract the exponents.

Show the answer

Answer: 3

  1. Simplify the numerator using the rule for multiplying exponents with the same base. The rule is: a^m × a^n = a^(m+n). Here, 3^2 × 3^3 = 3^(2+3) = 3^5. So the expression becomes: 3^5 ÷ 3^4.
  2. Simplify the division using the rule for dividing exponents with the same base. The rule is: a^m ÷ a^n = a^(m-n). Here, 3^5 ÷ 3^4 = 3^(5-4) = 3^1.
  3. Simplify 3^1. Any number raised to the power of 1 is the number itself. So 3^1 = 3. Final Answer: 3

3 (5³ × 5⁻²) ÷ 5² = ?

Hint: When multiplying terms with the same base, add the exponents. When dividing, subtract the exponents. Remember that a negative exponent indicates a reciprocal.

Show the answer

Answer: 0.2

  1. Apply the product of powers property to the numerator: 5³ × 5⁻² = 5^(3 + (-2)) = 5¹
  2. Now we have 5¹ ÷ 5²
  3. Apply the quotient of powers property: 5¹ ÷ 5² = 5^(1 - 2) = 5⁻¹
  4. Rewrite the negative exponent as a reciprocal: 5⁻¹ = 1/5¹ = 1/5
  5. Convert to decimal: 1/5 = 0.2

The answer is 0.2.

4 (2³ × 3²) ÷ (2² × 3) = ?

Hint: When simplifying expressions with exponents, remember that dividing powers with the same base means subtracting exponents. Also, any number to the power of zero equals one.

Show the answer

Answer: 6

  1. Write out the numerator and denominator clearly. Numerator: 2³ × 3² Denominator: 2² × 3 So the expression is: (2³ × 3²) / (2² × 3)
  2. Apply the laws of exponents. When dividing powers with the same base, we subtract exponents. For base 2: 2³ / 2² = 2^(3 - 2) = 2¹ = 2 For base 3: 3² / 3¹ = 3^(2 - 1) = 3¹ = 3
  3. Multiply the results from
  4. 2 × 3 = 6
  5. Conclusion. Therefore, (2³ × 3²) ÷ (2² × 3) = 6. Final answer: 6

Let's solve step-by-step. We have: (2³ × 3²) ÷ (2² × 3)

5 (5² × 2³) ÷ (5 × 2²) = ?

Hint: Apply the properties of exponents to simplify the expression before calculating. Remember that when dividing terms with the same base, you subtract exponents.

Show the answer

Answer: 10

  1. Write the expression: (5² × 2³) ÷ (5 × 2²)
  2. Apply the quotient rule for exponents: 5² ÷ 5¹ = 5^(2-1) = 5¹
  3. Apply the quotient rule for exponents: 2³ ÷ 2² = 2^(3-2) = 2¹
  4. The simplified expression is 5¹ × 2¹
  5. Calculate 5 × 2 = 10

The answer is 10.

6 (4² × 2³) ÷ (2⁴ × 4) = ?

Hint: Remember that when dividing exponential expressions with the same base, you subtract the exponents. Try rewriting all terms with the same base first.

Show the answer

Answer: 2

  1. Rewrite all terms with base 2. Note that 4 = 2², so 4² = (2²)² = 2⁴
  2. The expression becomes (2⁴ × 2³) ÷ (2⁴ × 2²)
  3. Simplify numerator: 2⁴ × 2³ = 2^(4+3) = 2⁷
  4. Simplify denominator: 2⁴ × 2² = 2^(4+2) = 2⁶
  5. Now we have 2⁷ ÷ 2⁶ = 2^(7-6) = 2¹
  6. 2¹ = 2

The answer is 2.

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