Example

Simplify: [latex]\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}[/latex]. Solution:

	[latex]\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}[/latex]
Simplify the numerator.	[latex]\frac{\frac{1}{4}}{4+{3}^{2}}[/latex]
Simplify the term with the exponent in the denominator.	[latex]\frac{\frac{1}{4}}{4+9}[/latex]
Add the terms in the denominator.	[latex]\frac{\frac{1}{4}}{13}[/latex]
Divide the numerator by the denominator.	[latex]\frac{1}{4}\div 13[/latex]
Rewrite as multiplication by the reciprocal.	[latex]\frac{1}{4}\cdot \frac{1}{13}[/latex]
Multiply.	[latex]\frac{1}{52}[/latex]

Example

Simplify: [latex]\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}[/latex].

Answer: Solution:

	[latex]\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}[/latex]
Rewrite numerator with the LCD of [latex]6[/latex] and denominator with LCD of [latex]12[/latex].	[latex]\frac{\frac{3}{6}+\frac{4}{6}}{\frac{9}{12}-\frac{2}{12}}[/latex]
Add in the numerator. Subtract in the denominator.	[latex]\frac{\frac{7}{6}}{\frac{7}{12}}[/latex]
Divide the numerator by the denominator.	[latex]\frac{7}{6}\div \frac{7}{12}[/latex]
Rewrite as multiplication by the reciprocal.	[latex]\frac{7}{6}\cdot \frac{12}{7}[/latex]
Rewrite, showing common factors.	[latex]\frac{\overline{)7}\cdot \overline{)6}\cdot 2}{\overline{)6}\cdot \overline{)7}\cdot 1}[/latex]
Simplify.	[latex]2[/latex]

Example

Evaluate [latex]x+\frac{1}{3}[/latex] when

1. [latex]x=-\frac{1}{3}[/latex]
2. [latex]x=-\frac{3}{4}[/latex]

Solution 1. To evaluate [latex]x+\frac{1}{3}[/latex] when [latex]x=-\frac{1}{3}[/latex], substitute [latex]-\frac{1}{3}[/latex] for [latex]x[/latex] in the expression.

	[latex]x+\frac{1}{3}[/latex]
Substitute [latex]\color{red}{--\frac{1}{3}}[/latex] for x.	[latex]\color{red}{--\frac{1}{3}} + \frac{1}{3}[/latex]
Simplify.	[latex]0[/latex]

2. To evaluate [latex]x+\frac{1}{3}[/latex] when [latex]x=-\frac{3}{4}[/latex], we substitute [latex]-\frac{3}{4}[/latex] for [latex]x[/latex] in the expression.

	[latex]x+\frac{1}{3}[/latex]
Substitute [latex]\color{red}{--\frac{3}{4}}[/latex] for x.	[latex]\color{red}{--\frac{3}{4}} + \frac{1}{3}[/latex]
Rewrite as equivalent fractions with the LCD, [latex]12[/latex].	[latex]-\frac{3\cdot 3}{4\cdot 3}+\frac{1\cdot 4}{3\cdot 4}[/latex]
Simplify the numerators and denominators.	[latex]-\frac{9}{12}+\frac{4}{12}[/latex]
Add.	[latex]-\frac{5}{12}[/latex]

Example

Evaluate [latex]y-\frac{5}{6}[/latex] when [latex]y=-\frac{2}{3}[/latex]

Answer: Solution: We substitute [latex]-\frac{2}{3}[/latex] for [latex]y[/latex] in the expression.

	[latex]y-\frac{5}{6}[/latex]
Substitute [latex]\color{red}{--\frac{2}{3}}[/latex] for y.	[latex]\color{red}{--\frac{2}{3}} - \frac{5}{6}[/latex]
Rewrite as equivalent fractions with the LCD, [latex]6[/latex].	[latex]-\frac{4}{6}-\frac{5}{6}[/latex]
Subtract.	[latex]-\frac{9}{6}[/latex]
Simplify.	[latex]-\frac{3}{2}[/latex]

Example

Evaluate [latex]2{x}^{2}y[/latex] when [latex]x=\frac{1}{4}[/latex] and [latex]y=-\frac{2}{3}[/latex]

Answer: Solution: Substitute the values into the expression. In [latex]2{x}^{2}y[/latex], the exponent applies only to [latex]x[/latex].

	[latex]2{x}^{2}y[/latex]
Substitute [latex]\color{red}{\frac{1}{4}}[/latex] for x and [latex]\color{blue}{--\frac{2}{3}}[/latex] for y.	[latex]2(\color{red}{\frac{1}{4}})^{2}(\color{blue}{--\frac{2}{3}})[/latex]
Simplify exponents first.	[latex]2(\frac{1}{16})(--\frac{2}{3})[/latex]
Multiply. The product will be negative.	[latex]--\frac{2}{1}\cdot\frac{1}{16}\cdot\frac{2}{3}[/latex]
Simplify.	[latex]--\frac{4}{48}[/latex]
Remove the common factors.	[latex]--\frac{1\cdot\color{red}{4}}{\color{red}{4} \cdot\ 12}[/latex]
Simplify.	[latex]--\frac{1}{12}[/latex]

Example

Evaluate [latex]\frac{p+q}{r}[/latex] when [latex]p=-4,q=-2[/latex], and [latex]r=8[/latex]

Answer: Solution: We substitute the values into the expression and simplify.

	[latex]\frac{p+q}{r}[/latex]
Substitute [latex]\color{red}{--4}[/latex] for p, [latex]\color{blue}{--2}[/latex] for q, and [latex]\color{red}{8}[/latex] for r.	[latex]\frac{\color{red}{--4} + \color{blue}{(--2)}}{\color{red}{8}}[/latex]
Add in the numerator first.	[latex]-\frac{6}{8}[/latex]
Simplify.	[latex]-\frac{3}{4}[/latex]