Problem 41 - Algebra for College Students

Question

Simplify. $$\frac{w^{-1}+y^{-1}}{z^{-1}+y^{-1}}$$

Step-by-Step Solution

Answer

\(\frac{z(y + w)}{w(y + z)}\)

Step 1: Rewrite Negative Exponents

Rewrite the negative exponents as their positive equivalents using the property \(\frac{1}{a^{-n}} = a^n\). The given expression becomes \(\frac{\frac{1}{w} + \frac{1}{y}}{\frac{1}{z} + \frac{1}{y}}\).

Step 2: Find Common Denominator in the Numerator

Find a common denominator for the terms in the numerator. The common denominator for \(\frac{1}{w}\) and \(\frac{1}{y}\) is \(wy\). Thus, rewrite the numerator: \(\frac{\frac{y + w}{wy}}{\frac{1}{z} + \frac{1}{y}}\).

Step 3: Find Common Denominator in the Denominator

Similarly, find a common denominator for \(\frac{1}{z}\) and \(\frac{1}{y}\). The common denominator for \(\frac{1}{z} \text{ and } \frac{1}{y}\) is \(zy\). Thus, rewrite the denominator: \(\frac{\frac{y + z}{zy}}{\frac{y + w}{wy}}\).

Step 4: Divide Fractions by Multiplying by the Reciprocal

Simplify the expression \(\frac{\frac{y + w}{wy}}{\frac{y + z}{zy}}\) by multiplying by the reciprocal of the denominator: \(\frac{y + w}{wy} \times \frac{zy}{y + z}\).

Step 5: Simplify the Expression

Cancel out common terms in the numerator and the denominator: \(\frac{y + w}{wy} \times \frac{zy}{y + z} = \frac{(y + w) \times zy}{wy \times (y + z)} = \frac{(y + w) \times z}{w \times (y + z)}\). Hence, the simplified form is \(\frac{z(y + w)}{w(y + z)}\).

Key Concepts

Negative Exponents

Let's start by understanding negative exponents. A negative exponent indicates that the base should be taken to the reciprocal and raised to the positive power. For example, if we have an exponent like \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). Thus, \(w^{-1}\) becomes \(\frac{1}{w}\) and \(y^{-1}\) becomes \(\frac{1}{y}\).
By applying this to the original expression \(\frac{w^{-1}+y^{-1}}{z^{-1}+y^{-1}}\), we get \(\frac{\frac{1}{w} + \frac{1}{y}}{\frac{1}{z} + \frac{1}{y}}\). Understanding this property helps in converting complex fractions into more manageable forms. This step is crucial because it sets the stage for further simplification by bringing all terms to a common ground: fractions with positive exponents.

Common Denominator

To combine fractions or simplify an expression involving fractions, finding a common denominator is essential. The common denominator for fractions identifies a common multiple for the denominators, allowing us to add or subtract the fractions easily. For example, if we have the fractions \(\frac{1}{w}\) and \(\frac{1}{y}\), their common denominator would be \(wy\).
We apply this concept in our numerator: \(\frac{\frac{1}{w} + \frac{1}{y}}{\frac{1}{z} + \frac{1}{y}}\) becomes \(\frac{\frac{y+w}{wy}}{\frac{1}{z} + \frac{1}{y}}\). This transformation helps to combine terms and prepare the expression for further simplification.
Similarly, in the denominator, the common denominator for \(\frac{1}{z}\) and \(\frac{1}{y}\) is \(zy\). So, \(\frac{\frac{1}{z} + \frac{1}{y}}\) becomes \(\frac{y+z}{zy}\). Again, this makes it easier to simplify the complex fraction by turning it into a simpler fraction with common denominators.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. For our transformed expression \(\frac{\frac{y+w}{wy}}{\frac{y+z}{zy}}\), we simplify by multiplying by the reciprocal of the denominator. The reciprocal of \(\frac{y+z}{zy}\) is \(\frac{zy}{y+z}\), so the expression becomes \(\frac{y+w}{wy} \times \frac{zy}{y+z}\).
To simplify, we can cancel out common factors in the numerator and denominator. Here, \(y\) and \(y\) in both the numerator and the denominator can be canceled, resulting in \(\frac{(y+w) \times z}{w \times (y+z)}\). This is our simplified expression.
Understanding how to simplify fractions helps in reducing complex expressions to their simplest forms, making calculations easier and more manageable. Always look for common terms to cancel out and ensure that the final expression is in its most reduced form.