Simplifying Exponential Expressions

Exponents and Exponential Expressions

Simplifying Exponential Expressions

Exponential expressions often involve applying the rules of exponents to simplify algebraic terms. Understanding these rules is essential for manipulating and simplifying expressions in precalculus.

• Exponent Rule (Power of a Power): When raising a power to another power, multiply the exponents: $(a^m)^n = a^{mn}$

• Exponent Rule (Power of a Product): When raising a product to a power, raise each factor to that power: $(ab)^n = a^n b^n$

Example: Simplify $(-2x^3y^2)^3$

• Step 1: Apply the power to each factor inside the parentheses:

  • $(-2x^3y^2)^3 = (-2)^3 (x^3)^3 (y^2)^3$

• Step 2: Simplify each term:

  • $(-2)^3 = -8$

  • $(x^3)^3 = x^{3 \times 3} = x^9$

  • $(y^2)^3 = y^{2 \times 3} = y^6$

• Step 3: Combine the results:

  • $(-2x^3y^2)^3 = -8x^9y^6$

Final Answer:

$(-2x^3y^2)^3 = -8x^9y^6$

Key Points to Remember

• Always apply the exponent to every factor inside the parentheses.

• Multiply exponents when raising a power to a power.

• Negative bases raised to an odd power remain negative.