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Laws of Exponents

Learn the rules for multiplying, dividing, and raising powers to powers

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Laws of Exponents

Product Rule

When multiplying powers with the same base, add exponents: am×an=am+na^m \times a^n = a^{m+n}

Example: x3×x4=x3+4=x7x^3 \times x^4 = x^{3+4} = x^7

Quotient Rule

When dividing powers with the same base, subtract exponents: aman=am−n\frac{a^m}{a^n} = a^{m-n}

Example: x7x3=x7−3=x4\frac{x^7}{x^3} = x^{7-3} = x^4

Power Rule

When raising a power to a power, multiply exponents: (am)n=amn(a^m)^n = a^{mn}

Example: (x3)4=x3×4=x12(x^3)^4 = x^{3 \times 4} = x^{12}

Zero Exponent

Any non-zero number to the zero power equals 1: a0=1(a≠0)a^0 = 1 \quad (a \neq 0)

Example: 50=15^0 = 1

Negative Exponent

A negative exponent means reciprocal: a−n=1ana^{-n} = \frac{1}{a^n}

Example: 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

📚 Practice Problems

1Problem 1easy

❓ Question:

Simplify: x5×x3x^5 \times x^3

💡 Show Solution

Solution:

Use the product rule (add exponents): x5×x3=x5+3=x8x^5 \times x^3 = x^{5+3} = x^8

Answer: x8x^8

2Problem 2easy

❓ Question:

Simplify: x5×x3x^5 \times x^3

💡 Show Solution

Solution:

Use the product rule (add exponents): x5×x3=x5+3=x8x^5 \times x^3 = x^{5+3} = x^8

Answer: x8x^8

3Problem 3medium

❓ Question:

Simplify: (x4)3x7\frac{(x^4)^3}{x^7}

💡 Show Solution

Solution:

Step 1: Power rule in numerator (x4)3=x4×3=x12(x^4)^3 = x^{4 \times 3} = x^{12}

Step 2: Quotient rule x12x7=x12−7=x5\frac{x^{12}}{x^7} = x^{12-7} = x^5

Answer: x5x^5

4Problem 4medium

❓ Question:

Simplify: (x4)3x7\frac{(x^4)^3}{x^7}

💡 Show Solution

Solution:

Step 1: Power rule in numerator (x4)3=x4×3=x12(x^4)^3 = x^{4 \times 3} = x^{12}

Step 2: Quotient rule x12x7=x12−7=x5\frac{x^{12}}{x^7} = x^{12-7} = x^5

Answer: x5x^5

5Problem 5hard

❓ Question:

Simplify: (2x3)44x5\frac{(2x^3)^4}{4x^5}

💡 Show Solution

Solution:

Step 1: Apply power to each factor in numerator (2x3)4=24×(x3)4=16x12(2x^3)^4 = 2^4 \times (x^3)^4 = 16x^{12}

Step 2: Divide 16x124x5=164×x12x5=4x7\frac{16x^{12}}{4x^5} = \frac{16}{4} \times \frac{x^{12}}{x^5} = 4x^7

Answer: 4x74x^7

6Problem 6hard

❓ Question:

Simplify: (2x3)44x5\frac{(2x^3)^4}{4x^5}

💡 Show Solution

Solution:

Step 1: Apply power to each factor in numerator (2x3)4=24×(x3)4=16x12(2x^3)^4 = 2^4 \times (x^3)^4 = 16x^{12}

Step 2: Divide 16x124x5=164×x12x5=4x7\frac{16x^{12}}{4x^5} = \frac{16}{4} \times \frac{x^{12}}{x^5} = 4x^7

Answer: 4x74x^7

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❓ Frequently Asked Questions

What is Laws of Exponents?▾
Learn the rules for multiplying, dividing, and raising powers to powers
How can I study Laws of Exponents effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 6 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Laws of Exponents study guide free?▾
Yes — all study notes, flashcards, and practice problems for Laws of Exponents on Study Mondo are free to access. No account is needed.
What course covers Laws of Exponents?▾
Laws of Exponents is part of the Grade 8 Math course on Study Mondo, specifically in the Exponents and Scientific Notation section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Laws of Exponents?▾
Yes, this page includes 6 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.