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Chapter 10: Problem 77

Factor each numerator and denominator. Then simplify if possible. \(\frac{2 x-14}{2}\)

Short Answer

Expert verified
The simplified form of the expression is \(x - 7\).

Step by step solution

01

Factor the Numerator

The numerator is \(2x - 14\). Notice that you can factor out a 2 from each term in the expression: \(2(x - 7)\). This is the factored form of the numerator.
02

Factor the Denominator

The denominator is already a simple number: \(2\). Since it is a single term, it doesn't need further factoring.
03

Simplify the Expression

With the factored numerator \(2(x - 7)\) and the denominator \(2\), you can simplify the fraction. The \(2\) in the numerator cancels with the \(2\) in the denominator, leaving you with \(x - 7\).
04

Final Simplified Form

After canceling out the common factor, the expression simplifies to \(x - 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplification
Simplification is a central process in math that involves transforming an expression into its simplest form. The aim is to remove complexity without changing the value. Simplification often occurs by factoring and canceling out common terms.
It makes math easier to understand and solve.
  • Identifying common factors is a key step in simplification.
  • After factoring, terms can be canceled, reducing the expression to a more manageable form.
In the provided exercise, we began by factoring the numerator, allowing us to easily simplify the fraction. This acts like cleaning up a math problem to make it clearer and easier to work with.
Numerator and Denominator
In fractions, the numerator is the top part, and the denominator is the bottom part. Understanding these components is crucial for operations involving algebraic fractions.
Each part plays a distinct role:
  • The numerator indicates how many parts we are considering.
  • The denominator shows into how many parts the whole is divided.
In the exercise, we had a numerator of \(2x - 14\) and a simple denominator of \(2\). By focusing separately on each, we could effectively simplify the fraction.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions instead of just numbers. They follow the same principles as regular fractions but can appear more complex.
Here's how to handle them:
  • Always start by factorizing both the numerator and the denominator if possible.
  • Look for common factors that can be canceled to simplify the expression.
In our exercise, simplifying the algebraic fraction \(\frac{2x - 14}{2}\) involved factoring the numerator and then canceling out the common factor to arrive at \(x - 7\). This approach helps solve even the most intricate algebraic fractions smoothly.

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Most popular questions from this chapter

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$g(7)$$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(x^{3} y^{2}\right)^{1 / 4}}{\left(x^{-5} y^{-1}\right)^{-1 / 2}} $$

Simplify each exponential expression. $$\frac{\left(2 a^{-1} b^{2}\right)^{3}}{\left(8 a^{2} b\right)^{-2}}$$

Factor the common factor from the given expression. $$ x^{-1 / 3} ; 5 x^{-1 / 3}+x^{2 / 3} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(m^{2} n\right)^{1 / 4}}{m^{-1 / 2} n^{5 / 8}} $$
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