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Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.
Simplify: $\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}.$
$\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}$ | |
Simplify the numerator. | $\frac{\frac{1}{4}}{4+{3}^{2}}$ |
Simplify the term with the exponent in the denominator. | $\frac{\frac{1}{4}}{4+9}$ |
Add the terms in the denominator. | $\frac{\frac{1}{4}}{13}$ |
Divide the numerator by the denominator. | $\frac{1}{4}\xf713$ |
Rewrite as multiplication by the reciprocal. | $\frac{1}{4}\xb7\frac{1}{13}$ |
Multiply. | $\frac{1}{52}$ |
Simplify: $\frac{{\left(\frac{1}{3}\right)}^{2}}{{2}^{3}+2}$ .
$\frac{1}{90}$
Simplify: $\frac{1+{4}^{2}}{{\left(\frac{1}{4}\right)}^{2}}$ .
272
Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}.$
$\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}$ | |
Rewrite numerator with the LCD of 6 and denominator with LCD of 12. | $\frac{\frac{3}{6}+\frac{4}{6}}{\frac{9}{12}-\frac{2}{12}}$ |
Add in the numerator. Subtract in the denominator. | $\frac{\phantom{\rule{0.2em}{0ex}}\frac{7}{6}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{7}{12}\phantom{\rule{0.2em}{0ex}}}$ |
Divide the numerator by the denominator. | $\frac{7}{6}\xf7\frac{7}{12}$ |
Rewrite as multiplication by the reciprocal. | $\frac{7}{6}\xb7\frac{12}{7}$ |
Rewrite, showing common factors. | $\frac{\overline{)7}\xb7\overline{)6}\xb72}{\overline{)6}\xb7\overline{)7}\xb71}$ |
Simplify. | 2 |
Simplify: $\frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\frac{1}{3}}$ .
2
Simplify: $\frac{\frac{2}{3}-\frac{1}{2}}{\frac{1}{4}+\frac{1}{3}}$ .
$\frac{2}{7}$
We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.
Evaluate
$x+\frac{1}{3}$ when
ⓐ To evaluate $x+\frac{1}{3}$ when $x=-\frac{1}{3},$ substitute $-\frac{1}{3}$ for $x$ in the expression.
$x+\frac{1}{3}$ | |
Simplify. | $0$ |
ⓑ To evaluate $x+\frac{1}{3}$ when $x=-\frac{3}{4},$ we substitute $-\frac{3}{4}$ for $x$ in the expression.
$x+\frac{1}{3}$ | |
Rewrite as equivalent fractions with the LCD, 12. | $-\frac{3\xb73}{4\xb73}+\frac{1\xb74}{3\xb74}$ |
Simplify the numerators and denominators. | $-\frac{9}{12}+\frac{4}{12}$ |
Add. | $-\frac{5}{12}$ |
Evaluate:
$x+\frac{3}{4}$ when
Evaluate:
$y+\frac{1}{2}$ when
Evaluate $y-\frac{5}{6}$ when $y=-\frac{2}{3}.$
We substitute $-\frac{2}{3}$ for $y$ in the expression.
$y-\frac{5}{6}$ | |
Rewrite as equivalent fractions with the LCD, 6. | $-\frac{4}{6}-\frac{5}{6}$ |
Subtract. | $-\frac{9}{6}$ |
Simplify. | $-\frac{3}{2}$ |
Evaluate: $y-\frac{1}{2}$ when $y=-\frac{1}{4}.$
$-\frac{3}{4}$
Evaluate: $x-\frac{3}{8}$ when $x=-\frac{5}{2}.$
$-\frac{23}{8}$
Evaluate $2{x}^{2}y$ when $x=\frac{1}{4}$ and $y=-\frac{2}{3}.$
Substitute the values into the expression. In $2{x}^{2}y,$ the exponent applies only to $x.$
Simplify exponents first. | |
Multiply. The product will be negative. | |
Simplify. | |
Remove the common factors. | |
Simplify. |
Evaluate. $3a{b}^{2}$ when $a=-\frac{2}{3}$ and $b=-\frac{1}{2}.$
$-\frac{1}{2}$
Evaluate. $4{c}^{3}d$ when $c=-\frac{1}{2}$ and $d=-\frac{4}{3}.$
$\frac{2}{3}$
Evaluate $\frac{p+q}{r}$ when $p=\mathrm{-4},q=\mathrm{-2},$ and $r=8.$
We substitute the values into the expression and simplify.
$\frac{p+q}{r}$ | |
Add in the numerator first. | $-\frac{6}{8}$ |
Simplify. | $-\frac{3}{4}$ |
Evaluate: $\frac{a+b}{c}$ when $a=\mathrm{-8},b=\mathrm{-7},\text{and}\phantom{\rule{0.2em}{0ex}}c=6.$
$-\frac{5}{2}$
Evaluate: $\frac{x+y}{z}$ when $x=9,y=\mathrm{-18},\text{and}\phantom{\rule{0.2em}{0ex}}z=\mathrm{-6}.$
$\frac{3}{2}$
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