Problem 53 - Algebra A Combined Function

Question

Simplify each of the following. First, write each expression with positive exponents. Then simplify the complex fraction. The first step has been completed for Exercise \(51 .\) $$ \frac{y^{-2}}{1-y^{-2}} $$

Step-by-Step Solution

Answer

\( \frac{1}{y^2 - 1} \)

Step 1: Rewrite with Positive Exponents

To remove the negative exponents, recall that a negative exponent indicates a reciprocal. Therefore, \( y^{-2} \) can be rewritten as \( \frac{1}{y^2} \). Let's rewrite the expression: \[ \frac{y^{-2}}{1-y^{-2}} = \frac{\frac{1}{y^2}}{1 - \frac{1}{y^2}} \]

Step 2: Simplify the Complex Fraction

To simplify this expression, we must combine the fractions in the denominator first. We express \(1\) as \(\frac{y^2}{y^2}\) so that both parts of the denominator share a common denominator. \[ 1 - \frac{1}{y^2} = \frac{y^2}{y^2} - \frac{1}{y^2} = \frac{y^2 - 1}{y^2} \]

Step 3: Simplify Further by Compound Fraction Rule

Now plug this into the numerator. We have: \[ \frac{\frac{1}{y^2}}{\frac{y^2 - 1}{y^2}} \] To simplify a fraction \( \frac{a/b}{c/d} \), multiply \(a\) by the reciprocal of \(c/d\). Thus: \[ \frac{\frac{1}{y^2}}{\frac{y^2 - 1}{y^2}} = \frac{1}{y^2} \times \frac{y^2}{y^2-1} = \frac{1}{y^2-1} \]

Step 4: Final Result

After applying all the steps and simplifying, the simplified expression is: \[ \frac{1}{y^2 - 1} \]

Key Concepts

Negative Exponents

In algebra, negative exponents might seem tricky at first glance, but they are relatively straightforward. A negative exponent such as \( y^{-2} \) indicates that instead of multiplying by the base, you take its reciprocal and then apply the positive exponent. Essentially, \( y^{-2} = \frac{1}{y^2} \).

Understanding this concept is crucial not only for simplifying expressions but also for other operations involving exponents. Here is a simple breakdown of steps to tackle negative exponents:

• Identify any negative exponents in your expression.
• Convert them into positive exponents by taking the reciprocal of the base.
• Replace the negative exponent section with its equivalent positive exponent form.

This transformation simplifies the handling of expressions and removes any confusion associated with negative signs.

Reciprocals

Reciprocals are integral to understanding and simplifying complex fractions, especially when involved with negative exponents. A reciprocal of a number \( n \) is simply \( \frac{1}{n} \). So when dealing with negative exponents, such as \( y^{-2} \), converting it into a positive exponent leads to its reciprocal: \( \frac{1}{y^2} \).

Reciprocals are handy in many mathematical situations involving division and fractions. Here’s a quick guide to working with reciprocals in fraction operations:

• When rewriting negative exponents, take the reciprocal of the base.
• In division scenarios (complex fractions), multiply by the reciprocal rather than directly dividing, i.e., the division \( \frac{a}{b} \div \frac{c}{d} \) becomes multiplication by the reciprocal \( \frac{a}{b} \times \frac{d}{c} \).

Using reciprocals simplifies operations within complex fractions, leading to clearer and simpler expressions.

Simplification Steps

Simplifying complex fractions involves a series of clear, logical steps that progress from rewriting expressions to combining and reducing them into their simplest form. Let's outline the simplification process for the given fraction \( \frac{y^{-2}}{1-y^{-2}} \):

• Rewrite negative exponents: Convert all negative exponents to their positive forms, recognizing them as reciprocals when necessary. This gives \( \frac{\frac{1}{y^2}}{1-\frac{1}{y^2}} \).
• Find a common denominator: When dealing with subtraction in fractions, adjust terms to have a common denominator. The expression becomes \( \frac{1}{y^2} \div (\frac{y^2}{y^2} - \frac{1}{y^2}) = \frac{1}{y^2} \div \frac{y^2 - 1}{y^2} \).
• Apply compound fraction rules: For the overall fraction, change division into multiplication using the reciprocal. So, \( \frac{\frac{1}{y^2}}{\frac{y^2 - 1}{y^2}} \) simplifies to \( \frac{1}{y^2} \times \frac{y^2}{y^2-1} \).
• Cancel common terms: In the context of complex fractions, once common terms are identified, cancel them out to achieve the simplest form, resulting in \( \frac{1}{y^2-1} \).

By methodically applying these steps, one can simplify even the most complex fractions with ease, paving the way for more advanced mathematics topics.