Express $\frac{2}{6 x}+\frac{2}{8}$ as a single fraction in its lowest terms.
Answer (2)
Simplify the fractions: 6 x 2 = 3 x 1 and 8 2 = 4 1 .
Find a common denominator: The common denominator for 3 x 1 and 4 1 is 12 x .
Rewrite the fractions with the common denominator: 3 x 1 = 12 x 4 and 4 1 = 12 x 3 x .
Add the fractions: 12 x 4 + 12 x 3 x = 12 x 4 + 3 x .
The expression 6 x 2 + 8 2 as a single fraction in its lowest terms is 12 x 4 + 3 x .
Explanation
Understanding the Problem We are given the expression 6 x 2 + 8 2 and our goal is to express it as a single fraction in its simplest form. To do this, we need to find a common denominator and combine the two fractions.
Simplifying Fractions First, let's simplify each fraction individually. We can simplify 6 x 2 by dividing both the numerator and the denominator by 2, which gives us 3 x 1 . Similarly, we can simplify 8 2 by dividing both the numerator and the denominator by 2, which gives us 4 1 . So now our expression is 3 x 1 + 4 1 .
Finding a Common Denominator To add these two fractions, we need to find a common denominator. The least common multiple of 3 x and 4 is 12 x . So, we want to rewrite each fraction with a denominator of 12 x . To do this, we multiply the first fraction, 3 x 1 , by 4 4 , which gives us 12 x 4 . We multiply the second fraction, 4 1 , by 3 x 3 x , which gives us 12 x 3 x . Now our expression is 12 x 4 + 12 x 3 x .
Adding the Fractions Now that we have a common denominator, we can add the two fractions: 12 x 4 + 12 x 3 x = 12 x 4 + 3 x .
Final Answer Finally, we check if the resulting fraction can be simplified further. The numerator is 4 + 3 x , and the denominator is 12 x . There are no common factors between 4 + 3 x and 12 x , so the fraction is already in its simplest form. Therefore, the expression 6 x 2 + 8 2 expressed as a single fraction in its lowest terms is 12 x 4 + 3 x .
Examples
Fractions are used in everyday life, such as when calculating proportions in recipes. For example, if a recipe calls for 6 2 of a cup of flour and 8 2 of a cup of sugar, expressing these as a single fraction helps to determine the total amount of dry ingredients needed. Simplifying and combining fractions allows for easier measurement and a better understanding of the ratios involved, ensuring the recipe turns out perfectly.
To combine 6 x 2 + 8 2 into a single fraction, first simplify each term. Then find the least common denominator ( 12 x ), rewrite the fractions, and sum them to obtain 12 x 4 + 3 x as the final result. This fraction is now in its simplest form.
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