Integer Exponents and Scientific Notation

Integer Exponents and Scientific Notation

Negative Integer Exponents

Negative exponents represent the reciprocal of the base raised to the corresponding positive exponent.

• Definition: For any nonzero real number a and positive integer n, $a^{-n} = \frac{1}{a^n}$

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

Review: Multiplying and Dividing Powers

When multiplying or dividing expressions with the same base, use the properties of exponents to simplify.

• Multiplication: $a^m \cdot a^n = a^{m+n}$

• Division: $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$

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

• Example:

  • Evaluate $2^3 \cdot 2^4$

  • $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

Rules for Exponents (Product and Power)

These rules help simplify expressions involving exponents:

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

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$, $a \neq 0$

• Power Rule: $(a^m)^n = a^{mn}$

Example: Evaluate $(\frac{2^3}{3^2})^{-2}$

• Step 1: Apply the negative exponent: $(\frac{2^3}{3^2})^{-2} = (\frac{3^2}{2^3})^{2}$

• Step 2: Apply the power to numerator and denominator: $= \frac{(3^2)^2}{(2^3)^2}$

• Step 3: Simplify exponents: $= \frac{3^{2 \cdot 2}}{2^{3 \cdot 2}} = \frac{3^4}{2^6}$

• Step 4: Calculate values: $= \frac{81}{64}$