Problem 52 - Algebra for College Students

Question

Simplify. $$\frac{4 x^{-3}-6 x^{-5}}{2 x^{-5}}$$

Step-by-Step Solution

Answer

The simplified form is \[2 x^{2} - 3\].

Step 1 - Simplify the Fraction Separately

Break up the given fraction into separate fractions: \[ \frac{4 x^{-3}}{2 x^{-5}} - \frac{6 x^{-5}}{2 x^{-5}} \]

Step 2 - Simplify Each Fraction

Simplify each fraction separately: \[\frac{4 x^{-3}}{2 x^{-5}} = 2 x^{2}\] and \[\frac{6 x^{-5}}{2 x^{-5}} = 3 \]

Step 3 - Combine the Simplified Terms

Put the simplified fractions together to find the final answer: \[ 2 x^{2} - 3 \]

Key Concepts

Fraction Simplification

Fraction simplification is a crucial skill in algebra. It involves breaking down complex fractions into simpler forms. This process often requires dividing the numerator and the denominator by their greatest common divisor (GCD). Separating a complex fraction into smaller, manageable parts can make simplification easier.

In our exercise, we first separated the fraction: \[ \frac{4 x^{-3}}{2 x^{-5}} - \frac{6 x^{-5}}{2 x^{-5}} \], then simplified each part separately, leading to, \[ 2 x^{2} - 3 \]. Breaking it down like this makes the problem more approachable.

Negative Exponents

Negative exponents can seem tricky, but they follow straightforward rules. A negative exponent indicates that the base should be taken to the reciprocal. This means that \[ x^{-n} = \frac{1}{x^n} \].

For example, in the term \[ x^{-3} \], this would be equivalent to \[ \frac{1}{x^3} \]. When dividing terms with exponents, you use the rule: \[ a^m / a^n = a^{m-n} \].

Applying this to our exercise: \[ \frac{4 x^{-3}}{2 x^{-5}} = 2 x^{2} \] and \[ \frac{6 x^{-5}}{2 x^{-5}} = 3 \]. The exponents are subtracted when dividing, simplifying the expressions effectively.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving terms like addition, subtraction, and multiplication. Simplifying polynomials means combining like terms to form a shorter, more manageable expression.

For instance, in our exercise, we ended up with \[ 2 x^{2} - 3 \] after fraction simplification and applying negative exponent rules. This final expression combines the simplified terms into a clean polynomial.

Understanding the components of polynomials is key to mastering algebraic manipulations, helping you to work through and simplify even complex polynomial expressions confidently.