Problem 72 Write each quotient in lowest te... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 8: Problem 72

Write each quotient in lowest terms. $$ \frac{4 \sqrt{6}+6}{10} $$

Short Answer

Expert verified

\[ \frac{2 \sqrt{6} + 3}{5} \]

Step by step solution

Write the original expression

The original expression that needs to be simplified is \ \[ \frac{4 \sqrt{6}+6}{10} \]

Factor out the common term in the numerator

Notice that the numerator, \ 4 \sqrt{6} + 6, has a common factor of 2. Factor out 2: \ \[ \frac{2(2 \sqrt{6} + 3)}{10} \]

Simplify the fraction

Divide both the numerator and the denominator by 2: \ \[ \frac{2(2 \sqrt{6} + 3)}{10} = \frac{(2 \sqrt{6} + 3)}{5} \]

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Key Concepts

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

Factoring

Factoring is a crucial concept in simplifying fractions. It involves breaking down a complex expression into products of simpler factors. Here, we start with the expression \[ 4 \sqrt{6} + 6 \]
When we observe this, we see that both terms, 4\sqrt{6} and 6, share a common factor of 2. This process helps in reducing the expression further.

To factor out the common term, we rewrite the expression by pulling out 2:
\[ 4 \sqrt{6} + 6 = 2(2 \sqrt{6} + 3) \]
Now, the numerator has been factored out to make further simplification easier. It's always easier to work with simpler expressions.

Numerator

The numerator is the top part of a fraction. In the expression \[ \frac{4 \sqrt{6} + 6}{10} \]
The numerator is \[4 \sqrt{6} + 6\].
The goal is to make it as simplified as possible.

Using factoring, we have rewritten the numerator:
\[ 4 \sqrt{6} + 6 = 2(2 \sqrt{6} + 3) \].
This step is vital for reducing fractions, as it allows us to cancel out common factors shared with the denominator.
By simplifying the numerator, we are one step closer to simplifying the entire fraction.

Denominator

The denominator is the bottom part of a fraction. In our example \[ \frac{4 \sqrt{6} + 6}{10} \]
The denominator is 10.

The importance of the denominator lies in its role in determining the overall value of the fraction. After factoring and rewriting the numerator, our expression looks like:
\[\frac{2(2 \sqrt{6} + 3)}{10}\]
By dividing both the numerator and the denominator by the greatest common factor, in this case, 2:
\[\frac{2(2 \sqrt{6} + 3)}{10} = \frac{2 \sqrt{6} + 3}{5}\],
Our fraction is now at its simplest form. Simplifying the denominator helped in reaching this reduced expression, keeping our calculations cleaner and more manageable.

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91影视

Write each quotient in lowest terms. $$ \frac{4 \sqrt{6}+6}{10} $$

Short Answer

Step by step solution

Write the original expression

Factor out the common term in the numerator

Simplify the fraction

Key Concepts

Factoring

Numerator

Denominator

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