SAT Math - Solving Rational and Fractional Functions

Simplify $\frac{1}{5}+\frac{3}{8}$

$\frac{23}{40}$

$\frac{4}{13}$

$\frac{6}{17}$

$\frac{4}{40}$

Explanation

To simplify this problem we need to find the least common denominator between the two fractions. To do this we look at 5 and at 8. The least common number between these two is 40.

In order to rewrite each fraction in terms of a denominator of 40 we need to muliple as follows:

$\frac{1}{5}*\frac{8}{8} + \frac{3}{8}* \frac{5}{5}$

we are able to mulitply by 8/8 and 5/5 because those fractions are really just 1 written in a different format.

Now using order of opperations we get the following

$\frac{8}{40}+\frac{15}{40}$

Now we have a common denominator and can do our addition to get the simplfied number:

$\frac{23}{40}$

Solve the following equation to find .

$\frac{3}{14} + \frac{2}{7} = x-\frac{1}{7}$

$x=\frac{9}{14}$

$x=\frac{3}{14}+\frac{3}{7}$

$x=\frac{5}{14}$

$x=\frac{3}{7}$

Explanation

In order to be able to find , we must first find the least common denominator. In this case, it is :

$\frac{2}{7}= \frac{2\times 2}{7\times 2}=\frac{4}{14}$

$\frac{1}{7}=\frac{1\times 2}{7\times 2}=\frac{2}{14}$

The equation can now be written as:

$\frac{3}{14} + \frac{4}{14} = x-\frac{2}{14}$

Solving for , we get:

$\frac{3}{14}+\frac{4}{14} + \frac{2}{14}=x$

$x=\frac{9}{14}$

Simplify the following:

$\frac{3}{x^2-1}+\frac{4}{x+1}$

$\frac{4x-1}{x^2-1}$

$\frac{2}{x-1}$

$\frac{4x+2}{x^2+1}$

$\frac{4x-2}{x^2-1}$

Explanation

To simplify the sum of the two fractions, we must find the common denominator.

Simplifying the denominator of the first fraction, we get

$\frac{3}{x^2-1}=\frac{3}{(x-1)(x+1)}$

because the denominator is a difference of two squares, which follows the form

Now, we can rewrite the sum as

$\frac{3}{(x+1)(x-1)}+\frac{4}{x+1}$

It is far easier to see the common denominator now:

$\frac{3}{(x-1)(x+1)}+\frac{4(x-1)}{(x-1)(x+1)}=\frac{3+4x-4}{(x-1)(x+1)}=\frac{4x-1}{x^2-1}$

Simplify:

$\left (8 x ^{3 } \right )^{-2}$

$\frac{1}{64x^{6}}$

$\frac{1}{6x^{6}}$

$\frac{1}{16x^{6}}$

$6x^{6}$

$8x^{6}$

Explanation

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the properties of exponents as follows:

$\left (8 x ^{3 } \right )^{-2} = \frac{1}{\left (8 x ^{3 } \right )^{ 2}} = \frac{1}{8^{ 2} \left ( x ^{3 } \right )^{ 2}} =\frac{1}{64 x ^{3 \cdot 2 } } =\frac{1}{64 x ^{6 } }$

If $f(x) = 2\sqrt{x}-\sqrt{x-3}$ , what is the value of ?

$2i\sqrt3 -i\sqrt6$

$i\sqrt6$

$-2i\sqrt3$

Explanation

Substitute the value of negative three as .

$f(-3) = 2\sqrt{-3}-\sqrt{(-3)-3}$

$2\sqrt{-3}-\sqrt{-6} = 2\sqrt{-1}\cdot \sqrt{3} - \sqrt{-1}\cdot \sqrt{6}$

The terms will be imaginary. We can factor out an $i = \sqrt{-1}$ out of the right side. Replace them with .

The answer is: $2i\sqrt3 -i\sqrt6$

Simplify:

$\frac{25st^2}{6s^3t^5}\div\frac{30s^{-9}}{45s^2t^7}$

$\frac{25s^9t^4}{4}$

$\frac{25}{4s^9t^4}$

$\frac{4s^9t^4}{25}$

$\frac{4}{25s^2t}$

Explanation

To divid fractions, you need to multiply the first fraction by the reciprocal of the second.

$(\frac{25st^2}{6s^3t^5})\div(\frac{30s^{-9}}{45s^2t^7})=(\frac{25st^2}{6s^3t^5})\times(\frac{45s^2t^7}{30s^{-9}})$

Now, multiply the numerators and denominators together, then simplify.

$(\frac{25st^2}{6s^3t^5})\times(\frac{45s^2t^7}{30s^{-9}})=\frac{1125s^3t^9}{180s^{-6}t^5}=\frac{25s^9t^4}{4}$

Solve:

$\frac{3}{x+1}+\frac{4}{x^2-1}=0$

$x=-\frac{1}{3}$

$x=-\frac{5}{3}$

$x=\frac{5}{3}$

Explanation

To solve this equation, we must find the least common denominator to add the fractions.

Keep in mind that the denominator of the second term is the difference of squares, which can be rewritten as

This is the least common denominator.

Now, we multiply both sides of the equationby the LCD on top and bottom (this is essentially 1):

$\frac{(x+1)(x-1)}{(x+1)(x-1)}\cdot (\frac{3}{x+1}+\frac{4}{x^2-1})=0$

After canceling terms, we get

$\frac{3(x-1)+4}{x^2-1}=0$

Now, we solve for x:

$x=-\frac{1}{3}$

Simplify:

$\frac{12m^2n}{16m^3n^4}\div\frac{8m^{-4}n}{18m^4n^2}$

$\frac{27m^7}{16n^2}$

$\frac{16n^2}{27m^7}$

$\frac{27n^2}{16m^7}$

Explanation

To divide fractions, you need to multiply the first fraction by the reciprocal of the second.

$(\frac{12m^2n}{16m^3n^4})\div(\frac{8m^{-4}n}{18m^4n^2})=(\frac{12m^2n}{16m^3n^4})\times(\frac{18m^4n^2}{8m^{-4}n})$

Now, multiply the numerators and denominators, then simplify.

$(\frac{12m^2n}{16m^3n^4})\times(\frac{18m^4n^2}{8m^{-4}n})=\frac{216m^6n^3}{128m^{-1}n^5}=\frac{27m^7}{16n^2}$

Solve: $\frac{1}{\sqrt{3x+1}} = \frac{1}{3}$

$\frac{8}{3}$

$\frac{8}{5}$

$\frac{1}{2}$

$\frac{1}{4}$

Explanation

Cross multiply to eliminate the fractions.

$(1)\sqrt{3x+1} = (1)(3)$

$\sqrt{3x+1} =3$

Square both sides.

$(\sqrt{3x+1})^2 =3^2$

Subtract 1 on both sides.

Divide by 3 on both sides.

$\frac{3x}{3}=\frac{8}{3}$

The answer is: $\frac{8}{3}$

Simplify $\frac{2}{3}+\frac{1}{4}+\frac{5}{6}$

$1\frac{3}{4}$

$\frac{8}{13}$

$\frac{11}{12}$

$1\frac{1}{6}$

$1\frac{1}{3}$

Explanation

Find the least common denominator (LCD) and convert each fraction to the LCD and then add. Simplify as necessary.

$\frac{2}{3}+\frac{1}{4}+\frac{5}{6}=\frac{8}{12}+\frac{3}{12}+\frac{10}{12}=\frac{21}{12}$

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

$\frac{21}{12}=\frac{7}{4}=1\frac{3}{4}$

Solving Rational and Fractional Functions

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