Problem 13 - Elementary Algebra

Question

For exercises 7-32, simplify. $$ \left(\frac{h^{2}}{h^{2}+3 h}\right)\left(\frac{h^{2}-9}{h}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(h - 3\).

Step 1: Factor the Expressions

Factor the denominators and numerators where possible. Notice that \(h^2 + 3h = h(h + 3)\) and \(h^2 - 9 = (h + 3)(h - 3)\).

Step 2: Rewrite the Expression with the Factors

Substitute these factors back into the original expression: \ \( \frac{h^2}{h(h + 3)} \times \frac{(h + 3)(h - 3)}{h} \).

Step 3: Simplify the Fractions

Simplify the fractions by canceling out common factors. The common factors \(h\) and \(h + 3\) appear in both the numerator and the denominator. This simplifies to: \ \( \frac{h \times h}{h \times (h + 3)} \times \frac{(h+3)(h-3)}{h} = \frac{h}{h + 3} \times (h - 3) \ = h - 3 \).

Step 4: Final Simplified Form

The expression simplifies to \(h - 3\).

Key Concepts

Factoring Expressions

Factoring is a key technique in algebra that breaks down complex expressions into simpler factors. Think of it as finding the building blocks of a number or expression. To factor an expression like \(h^2 + 3h\), look for common elements. Here, both terms have a common factor of \(h\), giving us \(h(h + 3)\).
Similarly, the expression \(h^2 - 9\) is a difference of squares, which factors to \((h + 3)(h - 3)\). Factoring expressions makes it easier to simplify fractions, set equations to zero, and solve for variables.

Canceling Common Factors

Once you've factored an expression, canceling common factors is the next step in simplification. Cancelling helps in reducing fractions by eliminating identical terms from the numerator and the denominator.
Take our example: after factoring, the expression is \( \frac{h^2}{h(h + 3)} \times \frac{(h + 3)(h - 3)}{h} \). Here, the common factors \(h\) and \(h + 3\) are present in both the numerator and the denominator. By canceling these out, the fraction becomes much simpler to work with.
Remember: you can only cancel terms that are multiplied together, not terms that are added or subtracted.

Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form. This process often includes factoring and canceling common factors. For our example, after factoring and canceling, we're left with \( \frac{h \times h}{h \times (h + 3)} \times \frac{(h+3)(h-3)}{h} \), which simplifies to \( \frac{h}{h + 3} \times (h - 3) = h - 3 \).
Each step in simplification requires careful attention to detail to ensure accuracy. Double-check your work: one small mistake can throw off the whole solution. Simplifying fractions makes them easier to interpret and use in further calculations or real-world problems.