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Exponents – Running Notes
17

Exponents (Level 1)

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Laws of Exponents

1. Product Law of Exponents

When the bases are the same and powers are multiplied, add the exponents.

For example:

$2^3 \times 2^5$

$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$= 2^8$

$= 2^{3+5}$

Therefore,

$a^m \times a^n = a^{m+n}$

Examples:

$2^5 \times 2^{10} = 2^{5+10} = 2^{15}$

$3^2 \times 3^7 = 3^{2+7} = 3^9$

$2^5 \times 2^{10} \times 2^{12} = 2^{5+10+12} = 2^{27}$

For three powers with the same base:

$a^m \times a^n \times a^p = a^{m+n+p}$

Example:

$5^{10} \times 5^2 \times 5^3 = 5^{15}$

2. Quotient Law of Exponents

When the bases are the same and powers are divided, subtract the exponents.

For example:

$\frac{2^8}{2^5}$

$= \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2 \times 2}$

After cancellation,

$= 2 \times 2 \times 2$

$= 2^3$

$= 2^{8-5}$

Therefore,

$\frac{a^m}{a^n} = a^{m-n}$

Examples:

$\frac{5^7}{5^2} = 5^{7-2} = 5^5$

$\frac{5^9}{5^3} = 5^{9-3} = 5^6$

3. Power of a Power Law

When a power is raised to another power, multiply the exponents.

For example:

$(5^3)^2 = 5^3 \times 5^3$

$= 5^{3+3}$

$= 5^6$

$= 5^{3(2)}$

Therefore,

$(a^m)^n = a^{mn}$

Examples:

$(7^5)^2 = 7^{5 \times 2} = 7^{10}$

$(6^3)^5 = 6^{3 \times 5} = 6^{15}$

$(7^2)^{10} = 7^{20}$

4. Power of Product Law

When a product is raised to a power, each factor gets the power.

For example:

$(xy)^3 = (xy)(xy)(xy)$

$= x \times x \times x \times y \times y \times y$

$= x^3 \times y^3 = x^3y^3$

Therefore,

$(ab)^n = a^n b^n$

Examples:

$(2x)^3 = 2^3 \cdot x^3 = 8x^3$

$(4y)^2 = 4^2 \cdot y^2 = 16y^2$

$(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6$

For three factors:

$(abc)^n = a^n b^n c^n$

Example:

$(2xy)^2 = 2^2 \cdot x^2 \cdot y^2 = 4x^2y^2$

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