Rational Expressions - GRE Quantitative Reasoning

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Question

Simplify the following rational expression: (9x - 2)/(x2) MINUS (6x - 8)/(x2)

Answer

Since both expressions have a common denominator, x2, we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

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Question

Simplify the following rational expression:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

Answer

Since both fractions in the expression have a common denominator of $x^{2}$ , we can combine like terms into a single numerator over the denominator:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

$=\frac{(7x-18)+(6x-14)}{x^{2}}$

$=\frac{13x-32}{x^{2}}$

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Question

Simplify the following rational expression:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

Answer

Since both rational terms in the expression have the common denominator $2x^{2}$ , combine the numerators and simplify like terms:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

$=\frac{(5x-5)+(7x+9)}{2x^{2}}$

$=\frac{12x+4}{2x^{2}}$

$=\frac{6x+2}{x^2}$

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Question

Simplify the following expression:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

Answer

Since both terms in the expression have the common denominator $x^{3}$ , combine the fractions and simplify the numerators:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

$=\frac{(10x-9)+(11x+12)}{x^{3}}$

$=\frac{21x+3}{x^{3}}$

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Question

Add and simplify:

$\frac{3x^2 + 3x + 3}{2y} + \frac{10x^2 + 3x}{2y}$

Answer

When adding rational expressions with common denominators, you simply need to add the like terms in the numerator.

Therefore, $\frac{13x^2 + 6x + 3}{2y}$ is the best answer.

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Question

Simplify the expression.

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

Answer

To add rational expressions, first find the least common denominator. Because the denominator of the first fraction factors to 2(x+2), it is clear that this is the common denominator. Therefore, multiply the numerator and denominator of the second fraction by 2.

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

$\frac{x}{2x+4}+\frac{(2)x}{(2)(x+2)}$

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

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

This is the most simplified version of the rational expression.

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Question

Simplify the following:

$\frac{x+3}{x-2}+\frac{x-2}{x+3}$

Answer

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

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

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

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Question

Choose the answer which best simplifies the following expression:

$\frac{3p}{2} + \frac{2p}{p^3}$

Answer

To simplify this expression, you have to get both numerators over a common denominator. The best way to go about doing so is to multiply both expressions by the others denominator over itself:

$\frac{3p}{2} \cdot \frac{p^3}{p^3 } + \frac{2p}{p^3} \cdot \frac{2}{2}$

Then you are left with:

$\frac{3p^4}{2p^3} + \frac{4p}{2p^3}$

Which you can simplify into:

$\frac{3p^4+4p}{2p^3}$

From there, you can take out a :

$\frac{p(3p^3 + 4)}{p(2p^2)}$

Which gives you your final answer:

$\frac{3p^3 +4}{2p^2}$

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Question

Choose the answer which best simplifies the following expression:

$\frac{2a + 2}{3} + \frac{3}{a^2}$

Answer

To solve this problem, first multiply both terms of the expression by the denominator of the other over itself:

$\frac{2a + 2}{3} \cdot \frac{a^2}{a^2} + \frac{3}{a^2} \cdot \frac{3}{3}$

$\frac{2a^3 +2a^2}{3a^2} + \frac{9}{3a^2}$

Now that both terms have a common denominator, you can add them together:

$\frac{2a^3 +2a^2 + 9}{3a^2}$

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Question

Choose the answer which best simplifies the following expression:

$\frac{3a^2 + a}{2} + \frac{10}{2a^2-a}$

Answer

To simplify, first multiply both terms by the denominator of the other term over itself:

$\frac{3a^2 + a}{2} \left(\frac{2a^2 - a}{2a^2 - a}\right) + \frac{10}{2a^2-a}\left(\frac{2}{2}\right)$

$\frac{6a^4 + 2a^3 -3a^3 - a^2}{4a^2 -2a} + \frac{20}{4a^2 - 2a}$

Then, you can combine the terms, now that they share a denominator:

$\frac{6a^4 - a^3 -a^2 +20}{4a^2 - 2a}$

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Question

Which of the following is equivalent to $\dpi{100} \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ ? Assume that denominators are always nonzero.

Answer

We will need to simplify the expression $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We can think of this as a large fraction with a numerator of $\frac{1}{t}-\frac{1}{x}$ and a denominator of $\dpi{100} x-t$ .

In order to simplify the numerator, we will need to combine the two fractions. When adding or subtracting fractions, we must have a common denominator. $\frac{1}{t}$ has a denominator of $\dpi{100} t$ , and $\dpi{100} -\frac{1}{x}$ has a denominator of $\dpi{100} x$ . The least common denominator that these two fractions have in common is $\dpi{100} xt$ . Thus, we are going to write equivalent fractions with denominators of $\dpi{100} xt$ .

In order to convert the fraction $\dpi{100} \frac{1}{t}$ to a denominator with $\dpi{100} xt$ , we will need to multiply the top and bottom by $\dpi{100} x$ .

$\frac{1}{t}=\frac{1\cdot x}{t\cdot x}=\frac{x}{xt}$

Similarly, we will multiply the top and bottom of $\dpi{100} -\frac{1}{x}$ by $\dpi{100} t$ .

$\frac{1}{x}=\frac{1\cdot t}{x\cdot t}=\frac{t}{xt}$

We can now rewrite $\frac{1}{t}-\frac{1}{x}$ as follows:

$\frac{1}{t}-\frac{1}{x}$ = $\frac{x}{xt}-\frac{t}{xt}=\frac{x-t}{xt}$

Let's go back to the original fraction $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We will now rewrite the numerator:

$\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ = $\frac{\frac{x-t}{xt}}{x-t}$

To simplify this further, we can think of $\frac{\frac{x-t}{xt}}{x-t}$ as the same as $\frac{x-t}{xt}\div (x-t)$ . When we divide a fraction by another quantity, this is the same as multiplying the fraction by the reciprocal of that quantity. In other words, $a\div b=a\cdot \frac{1}{b}$ .

$\frac{x-t}{xt}\div (x-t)$ = $\frac{x-t}{xt}\cdot \frac{1}{x-t}$ $=\frac{x-t}{xt(x-t)}$ $= \frac{1}{xt}$

Lastly, we will use the property of exponents which states that, in general, $\frac{1}{a}=a^{-1}$ .

$\frac{1}{xt}=(xt)^{-1}$

The answer is $(xt)^{-1}$ .

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Question

Simplify:

$\frac{x^{2}+x-2}{3x^{2} + 9x + 6} \div (x-1)$

Answer

Multiply by the reciprocal of $\frac{x-1}1{}$ .

$\frac{x^{2}+x-2}{3x^{2}+9x+6} \cdot \frac{1}{x-1}$

Factor $\frac{(x-1)(x+2)}{3(x+2)(x+1)} \cdot \frac{1}{x-1}$

Divide by common factors.

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

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Question

Solve for :

$\frac{\frac{2m^2 + 5}{4}}{ \frac{m}{2}} = 20$

Answer

To tackle this problem, you need to invert and multiply:

$\frac{\frac{2m^2 + 5}{4}} { \frac{m}{2}} = 20$

$\frac{2m^2 + 5}{4} \cdot \frac{2}{m} = 20$

$\frac{4m^2 + 10}{4m} = 20$

Here we see that we have created a quadratic equation. Therefore, we get all terms to one side, set it equal to zero and use the quadratic formula to solve.

The quadratic formula is:

$m=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where

Plugging these values in we get the following:

$\frac{80\pm\sqrt{80^2-4(4)(10)}}{2(4)}=\frac{80\pm\sqrt{6240}}{8}=\frac{80\pm78.99}{8}$

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Question

Express the following as a single rational expression:

$\frac{\frac{2x^2 + 7}{12b} }{ \frac{y^2 + 2y}{3a^2 + 2}}$

Answer

To divide one rational expression by another, invert and multiply:

$\frac{\frac{2x^2 + 7}{12b}}{ \frac{y^2 + 2y}{3a^2 + 2}}$

$\frac{2x^2 + 7 }{12b} \cdot \frac{3a^2 + 2}{y^2 + 2y}$

Remember to foil the numerator meaning, multiply the first components of each binomial. Then multiply the outer components of each binomial. After that, multiply the inner components together, and lastly, multiply the components in the last position of the binomials together.

This arrives at the following:

$\frac{6x^2a^2 +21a^2 +4x^2 +14}{12by^2 + 24by}$

You can't factor anything out, so that's your final answer.

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Question

If √(ab) = 8, and _a_2 = b, what is a?

Answer

If we plug in _a_2 for b in the radical expression, we get √(_a_3) = 8. This can be rewritten as a_3/2 = 8. Thus, log_a 8 = 3/2. Plugging in the answer choices gives 4 as the correct answer.

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Question

Answer

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Question

Find the product of and .

Answer

Solve the first equation for .

Solve the second equation for .

The final step is to multiply and .

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Question

Evaluate the following rational expression, if :

$\frac{2x^2 + 3}{x}$

Answer

To evaluate, simply plug in the number for :

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

Remembering to use order of operations (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) we arrive at our final solution.

$\frac{2(4)+3}{2}=\frac{8+3}{2}$

$=\frac{11}{2}$

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Question

If , find $\frac{2y}{y -8}$ .

Answer

To solve, simply plug in for :

$\frac{2y}{y -8}=$

Remembering to use the correct order of operations (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) we arrive at our final answer.

First do the multiplication that is in the numerator.

$\frac{2(3)}{3-8}=$

Now do the subtraction in the denominator.

$-\frac{6}{5}$

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Question

Find $\frac{2x^2 - 2x}{5}$ if .

Answer

To solve, simply plug in for :

$\frac{2x^2 - 2x}{5} =$

Remembering to use the correct order of operations (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) we arrive at the final solution.

$\frac{2(-1^2) - 2(-1))}{5} =$

Also recall that when a negative number is squared it becomes a positive number. This is also true when we multiply two negative numbers together.

$\frac{2(1)+1}{5}=$ $\frac{4}{5}$

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