Expressions - PSAT Math

Card 0 of 406

Question

Which of the following phrases can be written as the algebraic expression $\left (x - 19 \right )^{3}$ ?

Answer

$\left (x - 19 \right )^{3}$ is the cube of , which is the difference of a number and nineteen; therefore, $\left (x - 19 \right )^{3}$ is "the cube of the difference of a number and nineteen."

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Question

Which of the following phrases can be written as the algebraic expression $17 - y^{3}$ ?

Answer

$17 - y^{3}$ is seventeen decreased by $y^{3}$ , which is the cube of a number; therefore, $17 - y^{3}$ is "seventeen decreased by the cube of a number."

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Question

Increase by 70%. Which of the following will this be equal to?

Answer

A number increased by 70% is equivalent to 100% of the number plus 70% of the number. This is taking 170% of the number, or, equivalently, multiplying it by 1.7.

Therefore, increased by 70% is 1.7 times this, or

$1.7\left ( 6d - 13 \right ) = 1.7 \cdot 6d - 1.7 \cdot 13 = 10.2 d - 22.1$

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Question

Which of the following operations could represent the expression ?

Answer

Begin by putting the equation given into your own words. It might sound something similar to:

2 times x squared plus 7

Now, go through each answer choice and see if any of them are similar to this. We immediately see that the answer "7 more than 2 times the square of x" is similar to what we came up with. Let's do a quick run through of the other choices to be sure of our choice:

"2 times 7 more than the square of x" is equal to $\small 2(x^2+7)$ , which is equivalent to $\small 2x^2+14$ .

"2 times 7 less than the square of x" is equal to $\small \small 2(x^2-7)$ , which is equivalent to $\small \small 2x^2-14$ .

"7 more than the square of 2x" is equal to $\small (2x)^2+7$ , which is equivalent to $\small 4x^2+7$ .

"7 less than the square of 2x" is equal to $\small \small (2x)^2-7$ , which is equivalent to $\small \small 4x^2-7$ .

The only answer that works is "7 more than 2 times the square of x". This is the correct answer.

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Question

What does equal?

Answer

When solving a complex expression, remember the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). Reading left to right, begin by doing all operations within the innermost Parentheses first:

Continue simplifying using the acronym PEMDAS:

The expression is equal to -63.

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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

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

Answer

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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Question

what is 6/8 X 20/3

Answer

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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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

Which of the following phrases can be written as the algebraic expression $17 - y^{3}$ ?

Answer

$17 - y^{3}$ is seventeen decreased by $y^{3}$ , which is the cube of a number; therefore, $17 - y^{3}$ is "seventeen decreased by the cube of a number."

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Question

Increase by 70%. Which of the following will this be equal to?

Answer

A number increased by 70% is equivalent to 100% of the number plus 70% of the number. This is taking 170% of the number, or, equivalently, multiplying it by 1.7.

Therefore, increased by 70% is 1.7 times this, or

$1.7\left ( 6d - 13 \right ) = 1.7 \cdot 6d - 1.7 \cdot 13 = 10.2 d - 22.1$

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Question

Which of the following phrases can be written as the algebraic expression $\left (x - 19 \right )^{3}$ ?

Answer

$\left (x - 19 \right )^{3}$ is the cube of , which is the difference of a number and nineteen; therefore, $\left (x - 19 \right )^{3}$ is "the cube of the difference of a number and nineteen."

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Question

Which of the following operations could represent the expression ?

Answer

Begin by putting the equation given into your own words. It might sound something similar to:

2 times x squared plus 7

Now, go through each answer choice and see if any of them are similar to this. We immediately see that the answer "7 more than 2 times the square of x" is similar to what we came up with. Let's do a quick run through of the other choices to be sure of our choice:

"2 times 7 more than the square of x" is equal to $\small 2(x^2+7)$ , which is equivalent to $\small 2x^2+14$ .

"2 times 7 less than the square of x" is equal to $\small \small 2(x^2-7)$ , which is equivalent to $\small \small 2x^2-14$ .

"7 more than the square of 2x" is equal to $\small (2x)^2+7$ , which is equivalent to $\small 4x^2+7$ .

"7 less than the square of 2x" is equal to $\small \small (2x)^2-7$ , which is equivalent to $\small \small 4x^2-7$ .

The only answer that works is "7 more than 2 times the square of x". This is the correct answer.

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Question

What does equal?

Answer

When solving a complex expression, remember the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). Reading left to right, begin by doing all operations within the innermost Parentheses first:

Continue simplifying using the acronym PEMDAS:

The expression is equal to -63.

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Question

A rowing team paddles upstream at a rate of 10 miles every 2 hours and downstream at a rate of 27 miles every 3 hours. Assuming they are paddling at the same rate up and downstream, what is the speed of the water?

Answer

Upstream: p – w = (10/2) or p – w = 5 miles/hour

Downstream: p + w = (27/3) or p + w = 9 miles/hour

Then we add the two equations together to cancel out the w's. After adding we see

2p = 14

p = 7 miles/hour where p is the rate of the paddling. We plug p into the equation to find

w = 2 miles/hour where w is the rate of the stream's water.

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Question

If ab - bc + d = d2 - c2, then what is the value of a when b is two, c is negative one, and d is zero?

Answer

ab - bc + d = d2 - c2

We need to substitute values in for b, c, and d, and then solve the equation for a.

a(2) - 2(-1) + 0 = 02 - (-1)2

2a +2 + 0 = 0 - (1)

2a + 2 = -1

2a = -3

a = -3/2

The answer is -3/2.

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Question

If 11x + 4 = 19x – 12, then what is 2x – 4?

Answer

First solve for x. The first equation would simplify as:

16 = 8x

x = 2

If we plug x = 2 into the second expression:

2(2) – 4 = 0

0 is the correct answer.

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Question

If x = 2 and y = 3, then evaluate 2(x – 3) + 5y2

Answer

To evaluate an expression we make substitutions into the expression

2(x – 3) + 5y2 becomes 2(2 – 3) + 5 * 32 = –2 + 45 = 43

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Question

IF 5x3 = 40, then what is the value of 12x – (x/2)?

Answer

Use the first equation to solve for x, then plug into the 2nd equation to find a value.

5x3 = 40

x3 = 8

x = 2

12(2) – (2/2) = 24 – 1 = 23

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