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# what is the simplified form of the complex fraction?

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what is the simplified form of the complex fraction?

jbouyer  Nov 21, 2017

#2
+1493
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1)

$$\frac{\textcolor{red}{\frac{3}{x}+\frac{1}{4}}}{\textcolor{blue}{1+\frac{3}{x}}}$$ is quite the complex fraction. First, let's just deal with the numerator

 $$\textcolor{red}{\frac{3}{x}+\frac{1}{4}}$$ First. let's change this into fractions with common denominators. The LCM is 4x. $$\frac{12}{4x}+\frac{x}{4x}$$ Now, combine the fractions because we have formed a common denominator. $$\frac{12+x}{4x}$$

 $$\textcolor{blue}{1+\frac{3}{x}}$$ Just like before, transform the fractions to create a common denominator and combine. $$\frac{x}{x}+\frac{3}{x}$$ $$\frac{x+3}{x}$$

Now, write it as a fraction; you'll see how much easier it is to work with!

 $$\frac{\textcolor{red}{\frac{3}{x}+\frac{1}{4}}}{\textcolor{blue}{1+\frac{3}{x}}}=\frac{\frac{12+x}{4x}}{\frac{x+3}{x}}$$ $$\frac{\frac{12+x}{4x}}{\frac{x+3}{x}}$$ Multiply by $$\frac{x}{x+3}$$, the reciprocal of the complex denominator, to eliminate this complex fraction. $$\frac{x(12+x)}{4x(x+3)}$$ The x in the numerator and the x in the denominator cancel out here. $$\frac{12+x}{4(x+3)}$$ Now, distribute the 4 into every term. $$\frac{12+x}{4x+12}$$

2) $$\frac{c^2-4c+4}{12c^3+30c^2}\div\frac{c^2-4}{6c^4+15c^3}$$

Dividing by a fraction is the same as multiplying by its reciprocal.

$$\frac{c^2-4c+4}{12c^3+30c^2}*\frac{6c^4+15c^3}{c^2-4}$$

Now, let's factor the numerators and denominators completely and fully and see if any canceling can occur to simplify this.

 $$\frac{c^2-4c+4}{12c^3+30c^2}*\frac{6c^4+15c^3}{c^2-4}$$ Factor everything fully. $$\frac{(c-2)^2}{6c^2(2c+5)}*\frac{3c^3(2c+5)}{(c+2)(c-2)}$$ I see a lot of canceling that will occur here, dont you? $$\frac{c-2}{2}*\frac{c}{c+2}$$ Now, combine. $$\frac{c(c-2)}{2(c+2)}$$ This answer corresponds to the third one listed in the multiple guess.
TheXSquaredFactor  Nov 22, 2017
edited by TheXSquaredFactor  Nov 22, 2017
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#1
+1

X^2 answer is much simpler and easier to follow.

Guest Nov 22, 2017
edited by Guest  Nov 22, 2017
#2
+1493
+3

1)

$$\frac{\textcolor{red}{\frac{3}{x}+\frac{1}{4}}}{\textcolor{blue}{1+\frac{3}{x}}}$$ is quite the complex fraction. First, let's just deal with the numerator

 $$\textcolor{red}{\frac{3}{x}+\frac{1}{4}}$$ First. let's change this into fractions with common denominators. The LCM is 4x. $$\frac{12}{4x}+\frac{x}{4x}$$ Now, combine the fractions because we have formed a common denominator. $$\frac{12+x}{4x}$$

 $$\textcolor{blue}{1+\frac{3}{x}}$$ Just like before, transform the fractions to create a common denominator and combine. $$\frac{x}{x}+\frac{3}{x}$$ $$\frac{x+3}{x}$$

Now, write it as a fraction; you'll see how much easier it is to work with!

 $$\frac{\textcolor{red}{\frac{3}{x}+\frac{1}{4}}}{\textcolor{blue}{1+\frac{3}{x}}}=\frac{\frac{12+x}{4x}}{\frac{x+3}{x}}$$ $$\frac{\frac{12+x}{4x}}{\frac{x+3}{x}}$$ Multiply by $$\frac{x}{x+3}$$, the reciprocal of the complex denominator, to eliminate this complex fraction. $$\frac{x(12+x)}{4x(x+3)}$$ The x in the numerator and the x in the denominator cancel out here. $$\frac{12+x}{4(x+3)}$$ Now, distribute the 4 into every term. $$\frac{12+x}{4x+12}$$

2) $$\frac{c^2-4c+4}{12c^3+30c^2}\div\frac{c^2-4}{6c^4+15c^3}$$

Dividing by a fraction is the same as multiplying by its reciprocal.

$$\frac{c^2-4c+4}{12c^3+30c^2}*\frac{6c^4+15c^3}{c^2-4}$$

Now, let's factor the numerators and denominators completely and fully and see if any canceling can occur to simplify this.

 $$\frac{c^2-4c+4}{12c^3+30c^2}*\frac{6c^4+15c^3}{c^2-4}$$ Factor everything fully. $$\frac{(c-2)^2}{6c^2(2c+5)}*\frac{3c^3(2c+5)}{(c+2)(c-2)}$$ I see a lot of canceling that will occur here, dont you? $$\frac{c-2}{2}*\frac{c}{c+2}$$ Now, combine. $$\frac{c(c-2)}{2(c+2)}$$ This answer corresponds to the third one listed in the multiple guess.
TheXSquaredFactor  Nov 22, 2017
edited by TheXSquaredFactor  Nov 22, 2017

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