Section 1: Basic Rules of Exponents

Exponents are used to show repeated multiplication. For example, 4³ means 4 · 4 · 4 = 64.

In this section, we will review basic rules of exponents.

Product Rule of Exponents     a^maⁿ = a^{m + n}

When multiplying exponential expressions that have the same base, add the exponents.

Example:

Multiply: 4x³ • −6x2

Solution:

Multiply coefficients: 4 • −6 = −24

Use the product rule to multiply variables : x³ • x²  = x^{3 +2} =x5

4x³ · −6x² = −24x⁵

Quotient Rule of Exponents   

When dividing exponential expressions that have the same base, subtract the exponents.

Example:

Simplify: 

Solution:

Divide coefficients: 8 ÷ 2 = 4

Use the quotient rule to divide variables :  

Power Rule of Exponents     (a^m)ⁿ = a^mn

When raising an exponential expression to a new power, multiply the exponents.

Example:

Simplify: (7a⁴b⁶)²

Solution:

Each factor within the parentheses should be raised to the 2^nd power:

(7a⁴b⁶)² = 7²(a⁴)2(b⁶)²

Simplify using the Power Rule of Exponents :

(7a⁴b⁶)² = 7²(a⁴)²(b⁶)² = 49a⁸b¹²

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