### All PSAT Math Resources

## Example Questions

### Example Question #1 : How To Add Rational Expressions With A Common Denominator

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

**Possible Answers:**

**Correct answer:**

Since both expressions have a common denominator, x^{2}, 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.

### Example Question #1 : Rational Expressions

Simplify the expression.

**Possible Answers:**

**Correct 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.

This is the most simplified version of the rational expression.

### Example Question #1 : How To Evaluate Rational Expressions

If √(*ab*) = 8, and *a*^{2 }= *b*, what is *a*?

**Possible Answers:**

10

16

4

64

2

**Correct answer:**

4

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.

### Example Question #4 : Rational Expressions

**Possible Answers:**

–11/5

–37/15

9/5

37/15

–9/5

**Correct answer:**

–11/5

### Example Question #1 : How To Subtract Rational Expressions With Different Denominators

Simplify.

**Possible Answers:**

**Correct answer:**

Determine an LCD (Least Common Denominator) between and .

LCD =

Multiply the top and bottom of the first rational expression by , so that the denominator will be .

Distribute the to .

Now you can subtract because both rational expressions have the same denominators.

Final Answer.

### Example Question #1 : Rational Expressions

Which of the following is equivalent to ? Assume that denominators are always nonzero.

**Possible Answers:**

**Correct answer:**

We will need to simplify the expression . We can think of this as a large fraction with a numerator of and a denominator of .

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. has a denominator of , and has a denominator of . The least common denominator that these two fractions have in common is . Thus, we are going to write equivalent fractions with denominators of .

In order to convert the fraction to a denominator with , we will need to multiply the top and bottom by .

Similarly, we will multiply the top and bottom of by .

We can now rewrite as follows:

=

Let's go back to the original fraction . We will now rewrite the numerator:

=

To simplify this further, we can think of as the same as . 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, .

=

Lastly, we will use the property of exponents which states that, in general, .

The answer is .

### Example Question #1 : Rational Expressions

Simplify (4x)/(x^{2 }– 4) * (x + 2)/(x^{2 }– 2x)

**Possible Answers:**

x/(x + 2)

4/(x + 2)^{2}

4/(x – 2)^{2}

(4x^{2 }+ 8x)/(x^{4 }+ 8x)

x/(x – 2)^{2}

**Correct answer:**

4/(x – 2)^{2}

Factor first. The numerators will not factor, but the first denominator factors to (x – 2)(x + 2) and the second denomintaor factors to x(x – 2). Multiplying fractions does not require common denominators, so now look for common factors to divide out. There is a factor of x and a factor of (x + 2) that both divide out, leaving 4 in the numerator and two factors of (x – 2) in the denominator.

### Example Question #9 : Rational Expressions

what is 6/8 X 20/3

**Possible Answers:**

**Correct answer:**5

6/8 X 20/3 first step is to reduce 6/8 -> 3/4 (Divide top and bottom by 2)

3/4 X 20/3 (cross-cancel the threes and the 20 reduces to 5 and the 4 reduces to 1)

1/1 X 5/1 = 5

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