Properties of Exponents and Simplifying Expressions

Exponents and Their Properties

Understanding Exponents

Exponents are a fundamental concept in algebra, representing repeated multiplication of a base number. The rules for manipulating exponents are essential for simplifying algebraic expressions and solving equations.

• Exponentiation: The expression $a^n$ means multiplying the base a by itself n times.

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Power of a Power: $(a^m)^n = a^{mn}$

• Power of a Product: $(ab)^n = a^n b^n$

• Negative Exponents: $a^{-n} = \frac{1}{a^n}$

Simplifying the Given Expression

Let us simplify the expression:

$\left(a^{b+2}\right)^5 \left(a^{b-2}\right)^{-3}$

• Step 1: Apply the power of a power rule:

  • $\left(a^{b+2}\right)^5 = a^{5(b+2)} = a^{5b+10}$

  • $\left(a^{b-2}\right)^{-3} = a^{-3(b-2)} = a^{-3b+6}$

• Step 2: Multiply the results using the product of powers rule:

  • $a^{5b+10} \cdot a^{-3b+6} = a^{(5b+10)+(-3b+6)} = a^{2b+16}$

Final Simplified Form

$\left(a^{b+2}\right)^5 \left(a^{b-2}\right)^{-3} = a^{2b+16}$

Example Application

• If $a=2$ and $b=3$, then $a^{2b+16} = 2^{2\cdot3+16} = 2^{22}$.

Summary Table: Exponent Rules

Rule	Formula	Example
Product of Powers	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^4 = 2^{3+4} = 2^7$
Power of a Power	$(a^m)^n = a^{mn}$	$(3^2)^4 = 3^{2\cdot4} = 3^8$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$5^{-2} = \frac{1}{25}$