Multiplying Expressions with Exponents and Negative Powers

Exponents and Multiplication of Algebraic Expressions

Multiplying Expressions with Exponents

When multiplying algebraic expressions that include exponents, it is important to apply the properties of exponents and the rules for multiplying fractions.

• Product of Powers Property: When multiplying like bases, add the exponents: $a^m \cdot a^n = a^{m+n}$

• Multiplying Fractions: Multiply the numerators together and the denominators together.

Example Problem

Multiply the following expressions:

$\left( \frac{1}{9} x^{-2} y^7 \right) \left( \frac{1}{7} x^2 y^{-9} \right)$

• Step 1: Multiply the coefficients: $\frac{1}{9} \times \frac{1}{7} = \frac{1}{63}$

• Step 2: Apply the product of powers property to each variable:

  • $x^{-2} \times x^2 = x^{-2+2} = x^0 = 1$

  • $y^7 \times y^{-9} = y^{7+(-9)} = y^{-2}$

• Step 3: Combine all parts:

  • $\frac{1}{63} \cdot 1 \cdot y^{-2} = \frac{1}{63} y^{-2}$

• Step 4: Express with positive exponents (if required):

  • $y^{-2} = \frac{1}{y^2}$

  • So, $\frac{1}{63} y^{-2} = \frac{1}{63y^2}$

Final Answer:

$\left( \frac{1}{9} x^{-2} y^7 \right) \left( \frac{1}{7} x^2 y^{-9} \right) = \frac{1}{63y^2}$

Key Properties Used

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

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