ab/x²-16•x+4/a³b³
To simplify the expression \( \frac{ab}{x^2 - 16} \cdot \frac{x + 4}{a^3b^3} \), follow these steps:
Factor the Denominator: The expression \( x^2 - 16 \) is a difference of squares: \[ x^2 - 16 = (x - 4)(x + 4) \] Therefore, we can rewrite the expression as: \[ \frac{ab}{(x - 4)(x + 4)} \cdot \frac{x + 4}{a^3b^3} \]
Combine the Two Expressions: Now, multiply the two fractions: \[ \frac{ab(x + 4)}{(x - 4)(x + 4)} \cdot \frac{1}{a^3b^3} = \frac{ab(x + 4)}{(x - 4)(x + 4)a^3b^3} \]
Cancel Similar Terms: Notice that \( x + 4 \) appears in both the numerator and denominator: \[ \frac{ab}{(x - 4)a^3b^3} \]
Combine the Terms: This can be simplified further by reducing the coefficients: \[ \frac{1}{(x - 4)a^2b^2} \]
Final Result: Therefore, the simplified expression is: \[ \frac{1}{(x - 4)a^2b^2} \]
Keep in mind, the simplification above is valid as long as \( x \neq 4 \) and \( a \neq 0 \) and \( b \neq 0 \) (to avoid division by zero).
