How to evaluate rational expressions - GRE Quantitative Reasoning

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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

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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Question

Evaluate $\frac{x^2 - 3x}{x + 2}$ if .

Answer

To evaluate, merely plug in for :

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

$\frac{4^2 - (3\cdot4)}{4 + 2} =$

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

$\frac{16-(12)}{6}=\frac{4}{6}$

From here we can further reduce the fraction by factoring out a two from both the numerator and denominator.

$\frac{4}{6} =\frac{2\cdot 2}{2\cdot 3}$

Canceling out the two's we get:

$\frac{2}{3}$

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Question

Evaluate $\frac{y^3 - 3}{y - 3}$ if .

Answer

To solve, simply plug in for :

$\frac{y^3 - 3}{y - 3}$

$\frac{0^3 - 3}{0 - 3}$

Recall that when a negative number is divided by another negative number it results in a positive number.

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

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Question

Evaluate $\frac{x^2 + 2x - 2}{x}$ if .

Answer

To evaluate, simply plug in for :

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

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

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

$=\frac{25+10-2}{5}=\frac{35-2}{5}=\frac{33}{5}$

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Question

Evaluate $\frac{5n^2}{-2n - 8}$ if .

Answer

To solve, simply plug in for :

$\frac{5n^2}{-2n - 8}$

$\frac{5(-2)^2}{(-2\cdot-2)-8}$

Remembering the order of operations (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) we arrive at our final solution. Also recall that multiplying two negative numbers together leads to a positive product; this is also true when you square a negative number.

$\frac{5(4)}{4-8}=\frac{20}{-4}$

From here we can reduce the fraction by factoring out a four from both the numerator and the denominator.

$\frac{4\cdot 5}{4 \cdot -1}=-\frac{5}{1}=-5$

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Question

Evaluate $\frac{z^2 + 2z - 20}{3}$ if .

Answer

To evaluate, simply plug in for :

$\frac{z^2 + 2z - 20}{3}$

$\frac{3^2 + (2\cdot 3)-20}{3}$

Remembering the order of operations we are able to solve this problem. The order of operations is parentheses, exponents, multiplication, division, addition, subtraction.

$\frac{9 + (6)-20}{3}=\frac{15-20}{3}=-\frac{5}{3}$

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Question

Evaluate $\frac{x^3 + x^2}{x}$ if .

Answer

To evaluate, simply plug in for :

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

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

Recall that we must simplify the exponents before we are able to add the numerator and divide by the denominator.

$\frac{8 + 4}{2}=\frac{12}{2}$

From here we can reduce the fraction by factoring out a two from both the numerator and the denominator.

$\frac{12}{2}=\frac{2\cdot 6}{2\cdot 1}=\frac{6}{1}=6$

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Question

Evaluate $\frac{3x^2 + 9x -18}{2x^2 + x}$ if .

Answer

To solve, simply plug in for :

$\frac{3x^2 + 9x -18}{2x^2 + x}$

$\frac{3(-1)^2 + 9(-1) -18}{2(-1)^2 + -1}$

According to the order of operations we must simplify the exponents first. Then we will multiply the necessary terms and finally we will add and subtract the terms as specified.

$=\frac{3(1)-9-18}{2(1)-1}=\frac{3 -27}{2-1}$

$=\frac{-24}{1}=-24$

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Question

Evaluate $\frac{7y^2 - 8y}{y + 12}$ if .

Answer

To solve, simply plug in for :

$\frac{7y^2 - 8y}{y + 12}$

$\frac{7(4^2) - 8(4)}{4 + 12}$

First we must simplify the exponents. The we will multiply each of the two terms in the numerator. After that, we can add the terms in denominator and subtract the terms in the numerator.

$\frac{7(16)-32}{16}=\frac{112-32}{16}$

Sutracting the terms on top gives:

$\frac{80}{16}$ .

We can simplify this fraction by factoring out a sixteen from both the numerator and denominator.

$\frac{80}{16}=\frac{16\cdot 5}{16 \cdot 1}=\frac{5}{1}=5$

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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.

Compare your answer with the correct one above

Question

Find the product of and .

Answer

Solve the first equation for .

Solve the second equation for .

The final step is to multiply and .

Compare your answer with the correct one above

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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