Problem 110 - Elementary Algebra

Question

For exercises 97-114, evaluate. $$ \left(-\frac{3}{4}\right)^{2}+\frac{1}{2}-\frac{5}{8} $$

Step-by-Step Solution

Answer

\(\frac{7}{16}\)

Step 1 - Evaluate the Exponent

Calculate \( \left(-\frac{3}{4}\right)^{2} \). Since the exponent is 2, multiply \(-\frac{3}{4}\cdot-\frac{3}{4}\) resulting in \(\frac{9}{16}\).

Step 2 - Simplify the Expression

Combine all terms: \(\frac{9}{16}+\frac{1}{2}-\frac{5}{8}\).

Step 3 - Find a Common Denominator

The common denominator for 16, 2, and 8 is 16. Thus, express all fractions with the denominator 16: \(\frac{9}{16}+\frac{8}{16}-\frac{10}{16}\).

Step 4 - Add and Subtract the Fractions

Perform the addition and subtraction: \(\frac{9}{16}+\frac{8}{16}-\frac{10}{16}=\frac{7}{16}\).

Key Concepts

exponents

When we talk about exponents, we refer to a value that indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \((-3/4)^{2}\), the exponent is 2. This means we should multiply \((-3/4)\) by itself. It's important to recognize that multiplying two negative numbers results in a positive number:\((-3/4) \cdot (-3/4) = 9/16\).
So, \((- \frac{3}{4})^{2}= \frac{9}{16}\).
Understanding exponents and their rules helps simplify many algebraic expressions and solve complex problems accurately.

fractions

Fractions represent a part of a whole. They consist of a numerator and a denominator. The numerator sits above the fraction line and shows how many parts we have, while the denominator sits below and shows the total number of equal parts into which the whole is divided.
For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
Fractions require particular attention due to their different operations, like addition and subtraction, which need a common denominator to proceed.

common denominator

To add or subtract fractions, they must have a common denominator, meaning the denominators must be the same across all fractions. Finding a common denominator requires understanding the least common multiple (LCM) of the involved denominators.
For the expression \(\frac{9}{16}+\frac{1}{2}-\frac{5}{8}\), the denominators are 16, 2, and 8. The LCM of these numbers is 16, which is the smallest number into which all denominators can evenly divide. Therefore, we convert all fractions:

-\frac{9}{16}, -\frac{8}{16}, and -\frac{10}{16}(-5/8).

addition and subtraction of fractions

Once fractions have a common denominator, you can add or subtract them swiftly.
Consider the expression \(\frac{9}{16}+\frac{8}{16}-\frac{10}{16}\). After you've converted each fraction to have a denominator of 16, the next step is simple. Combine the numerators and keep the common denominator:
\(\frac{9}{16}+\frac{8}{16} = \frac{17}{16}\)
Remember, when dealing with negative numbers and fractions, always pay close attention to signs during addition and subtraction to avoid mistakes.