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PSAT Math : PSAT Mathematics

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Example Questions

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

Simplify the expression.

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

Possible Answers:

$\frac{2x}{3x+6}$

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

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

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

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

Correct answer:

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

Explanation:

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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Example Question #11 : Rational Expressions

If √(ab) = 8, and a²= b, what is a?

Possible Answers:

Correct answer:

Explanation:

If we plug in a² for b in the radical expression, we get √(a³) = 8. This can be rewritten as a^3/2 = 8. Thus, log_a8 = 3/2. Plugging in the answer choices gives 4 as the correct answer.

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Example Question #433 : Algebra

Function_part1

Possible Answers:

9/5

37/15

–11/5

–9/5

–37/15

Correct answer:

–11/5

Explanation:

Fraction_part2

Fraction_part3

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

Simplify.

$\frac{x+5}{x}-\frac{2x-3}{x^{2}}$

Possible Answers:

7x+3

3x+3

x+6

$\frac{x^{2}+7x+3}{x^{2}}$

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

Correct answer:

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

Explanation:

$\frac{x+5}{x}-\frac{2x-3}{x^{2}}$

Determine an LCD (Least Common Denominator) between and $x^{2}$ .

LCD = $x^{2}$

$\frac{x(x+5)}{(x)x}-\frac{2x-3}{x^{2}}$

Multiply the top and bottom of the first rational expression by , so that the denominator will be $x^{2}$ .

Distribute the to x+5 .

$\frac{x^{2}+5x}{x^{2}}-\frac{2x-3}{x^{2}}$

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

$\frac{x^{2}+5x-2x+3}{x^{2}}$

Final Answer.

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

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Example Question #1 : Expressions

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

Possible Answers:

$t-x$

$\frac{x}{t}$

$(xt)^{-1}$

$x^{2}-t^{2}$

$x-t$

Correct answer:

$(xt)^{-1}$

Explanation:

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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Example Question #1 : How To Multiply Rational Expressions

Simplify (4x)/(x²– 4) * (x + 2)/(x²– 2x)

Possible Answers:

(4x²+ 8x)/(x⁴+ 8x)

x/(x – 2)²

4/(x + 2)²

x/(x + 2)

4/(x – 2)²

Correct answer:

4/(x – 2)²

Explanation:

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.

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Example Question #4 : Rational Expressions

what is 6/8 X 20/3

Possible Answers:

3/20

9/40

120/11

18/160

Correct answer: 5

Explanation:

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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Example Question #1 : How To Add Exponents

If a² = 35 and b² = 52 then a⁴ + b⁶ = ?

Possible Answers:

522

140,608

3929

150,000

141,833

Correct answer:

141,833

Explanation:

a⁴= a² * a² and b⁶= b² * b²* b²

Therefore a⁴ + b⁶ = 35 * 35 + 52 * 52 * 52 = 1,225 + 140,608 = 141,833

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Example Question #1 : How To Add Exponents

If $9^{(x + 5)}+3^{2(x+5)}=162$ , what is the value of ?

Possible Answers:

Correct answer:

Explanation:

Since we have two ’s in $9^{(x + 5)} + 3^{2(x + 5)}$ we will need to combine the two terms.

For $3^{2(x + 5) }$ this can be rewritten as

$(3^2)^{ (x + 5)} = 9^ {(x + 5)}$

So we have $9^{ (x + 5) }+ 9^{ (x + 5)} = 162$ .

Or $2 (9^{ (x + 5)}) = 162$

Divide this by : $9^{ (x + 5)} = 81 = 9^ 2$

Thus x +5 = 2 or x = -3

*Hint: If you are really unsure, you could have plugged in the numbers and found that the first choice worked in the equation.

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Example Question #2 : How To Add Exponents

Solve for x.

2³+ 2^x+1= 72

Possible Answers:

Correct answer:

Explanation:

The answer is 5.

8 + 2^x+1= 72

2^x+1= 64

2^x+1= 2⁶

x + 1 = 6

x = 5

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PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

Example Question #11 : Rational Expressions

Example Question #433 : Algebra

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

Example Question #1 : Expressions

Example Question #1 : How To Multiply Rational Expressions

Example Question #4 : Rational Expressions

Example Question #1 : How To Add Exponents

Example Question #1 : How To Add Exponents

Example Question #2 : How To Add Exponents

All PSAT Math Resources

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