Laws of Exponents

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

Product Rule

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

Example: $x^3 \times x^4 = x^{3+4} = x^7$

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

Example: $\frac{x^7}{x^3} = x^{7-3} = x^4$

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

Example: $(x^3)^4 = x^{3 \times 4} = x^{12}$

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

Example: $5^0 = 1$

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

Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Simplify: $x^5 \times x^3$

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Solution:

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

Answer: $x^8$

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

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Solution:

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

Step 2: Quotient rule $\frac{x^{12}}{x^7} = x^{12-7} = x^5$

Answer: $x^5$

Simplify: $\frac{(2x^3)^4}{4x^5}$

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Solution:

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

Step 2: Divide $\frac{16x^{12}}{4x^5} = \frac{16}{4} \times \frac{x^{12}}{x^5} = 4x^7$

Answer: $4x^7$

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