### All ACT Math Resources

## Example Questions

### Example Question #403 : Algebra

You are told that can be determined from the expression:

Determine whether the absolute value of is greater than or less than 2.

**Possible Answers:**

The quantities are equal

The relationship cannot be determined from the information given.

**Correct answer:**

The expression is simplified as follows:

Since the value of must be slightly greater for it to be 17 when raised to the 4th power.

### Example Question #1 : How To Simplify An Expression

Which best describes the relationship between and if ?

**Possible Answers:**

The relationship cannot be determined from the information given.

**Correct answer:**

The relationship cannot be determined from the information given.

Use substitution to determine the relationship.

For example, we could plug in and .

So far it looks like the first expression is greater, but it's a good idea to try other values of x and y to be sure. This time, we'll try some negative values, say, and .

This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.

### Example Question #401 : Gre Quantitative Reasoning

If and , then

**Possible Answers:**

Cannot be determined

**Correct answer:**

We have three variables and only two equations, so we will not be able to solve for each independent variable. We need to think of another solution.

Notice what happens if we line up the two equations and add them together.

*(x* + *y*) + (3*x –* *y* + *z) *= 4x + z

and 5 + 3 = 8

Lets take this equation and multiply the whole thing by 3:

3(4*x* + *z* = 8)

Thus, 12*x* + 3*z* = 24.

### Example Question #5 : How To Simplify Expressions

Simplify the following expression: x^{3} - 4(x^{2} + 3) + 15

**Possible Answers:**

**Correct answer:**

To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x^{2 }and -4 by 3.

x^{3} - 4x^{2} -12 + 15

You can then add -12 and 15, which equals 3.

You now have x^{3} - 4x^{2} + 3 and are finished. Just a reminder that x^{3} and 4x^{2 }are not like terms as the x’s have different exponents.

### Example Question #25 : Simplifying Expressions

Which of the following is equivalent to ?

**Possible Answers:**

ab/c

a^{2}/(b^{5}c)

b^{5}/(ac)

abc

ab^{5}c

**Correct answer:**

b^{5}/(ac)

First, we can use the property of exponents that x^{y}/x^{z} = x^{y–z}

Then we can use the property of exponents that states x^{–y} = 1/x^{y}

a^{–1}b^{5}c^{–1 }= b^{5}/ac

### Example Question #26 : Simplifying Expressions

Simplify the following expression:

2x(x^{2} + 4ax – 3a^{2}) – 4a^{2}(4x + 3a)

**Possible Answers:**

–12a^{3 –} 14a^{2}x + 2x^{3}

12a^{3 }– 16a^{2}x + 8ax^{2} + 2x^{3}

–12a^{3 }– 14ax^{2} + 2x^{3}

–12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

**Correct answer:**

–12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

Begin by distributing each part:

2x(x^{2} + 4ax – 3a^{2}) = 2x * x^{2} + 2x * 4ax – 2x * 3a^{2} = 2x^{3} + 8ax^{2} – 6a^{2}x

The second:

–4a^{2}(4x + 3a) = –16a^{2}x – 12a^{3}

Now, combine these:

2x^{3} + 8ax^{2} – 6a^{2}x – 16a^{2}x – 12a^{3}

The only common terms are those with a^{2}x; therefore, this reduces to

2x^{3} + 8ax^{2} – 22a^{2}x – 12a^{3}

This is the same as the given answer:

–12a^{3 }– 22a^{2}x + 8ax^{2} + 2x^{3}

### Example Question #27 : Simplifying Expressions

Simplify the following expression:

(*xy*)^{2} – *x*((4*x*)(*y*)^{2 }– (4*x*)^{2}) – 4^{2}*x*^{2}

**Possible Answers:**

–3*xy*^{2 }– 4*x*^{3 }– 16*x*^{2}

–3*x*^{2}*y*^{2 }– 16*x*^{3} – 16*x*^{2}

5*x*^{2}*y*^{2}

–3*xy*^{2 }– 4*x*^{3}

3*x*^{2}*y*^{2 }+ 16*x*^{3} – 16*x*^{2}

**Correct answer:**

–3*x*^{2}*y*^{2 }– 16*x*^{3} – 16*x*^{2}

To simplify this, we will want to use the correct order of operations. The mnemonic device PEMDAS is usually very helpful.

Parenthesis (1st)

Exponents (2nd)

Multiply, Divide (3rd)

Add, Subtract (4th)

PEMDAS tells us to evaluate parentheses first, then exponents. After exponents, we evaluate multiplication and division from left to right, and then we evaluate addition and subtraction from left to right.

Let's look at (*xy*)^{2} – *x*((4*x*)(*y*)^{2 }– (4*x*)^{2}) – 4^{2}*x*^{2}

We want to start with parentheses first. We will simplify the (*xy*)^{2} by using the general rule of exponents, which states that (*ab*)* ^{c}* =

*a*. Thus we can replace (

^{c}b^{c}*xy*)

^{2}with

*x*

^{2}

*y*

^{2}.

*x*^{2}*y*^{2} – *x*((4*x*)(*y*)^{2 }– (4*x*)^{2}) – 4^{2}*x*^{2}

When we have parentheses within parentheses, we want to move from the innermost parentheses to the outermost. This means we will want to simplify the expression (4*x*)(*y*)^{2 }– (4*x*)^{2} first, which becomes 4*xy*^{2} – 16*x*^{2}. We can now replace (4*x*)(*y*)^{2 }– (4*x*)^{2} with 4*xy*^{2} – 16*x*^{2 }.

*x*^{2}*y*^{2} – *x*(4*xy*^{2} – 16*x*^{2}) – 4^{2}*x*^{2}

In order to remove the last set of parentheses, we will need to distribute the *x* to 4*xy*^{2} – 16*x*^{2} . We will also make use of the property of exponents which states that *a ^{b}a^{c}* =

*a*

^{b+c}.

*x*^{2}*y*^{2} – *x*(4*xy*^{2}) – *x*(16*x*^{2 }) – 4^{2}*x*^{2 }= *x*^{2}*y*^{2} – 4*x*^{2}*y*^{2 }– 16*x*^{3 }– 4^{2}*x*^{2}

We now have the parentheses out of the way. We must now move on to the exponents. Really, the only exponent we need to simplify is –4^{2}, which is equal to –16. Remember that –4^{2} = –(4^{2}), which is not the same as (–4)^{2}.

*x*^{2}*y*^{2} – 4*x*^{2}*y*^{2 }– 16*x*^{3} – 16*x*^{2}

Now, we want to use addition and subtraction. We need to add or subtract any like terms. The only like terms we have are *x*^{2}*y*^{2} and –4*x*^{2}*y*^{2}. When we combine those, we get –3*x*^{2}*y*^{2}

–3*x*^{2}*y*^{2 }– 16*x*^{3} – 16*x*^{2}

The answer is –3*x*^{2}*y*^{2 }– 16*x*^{3} – 16*x*^{2} .

### Example Question #28 : Simplifying Expressions

If both and are positive, what is the simplest form of ?

**Possible Answers:**

**Correct answer:**

can also be expressed as

### Example Question #21 : Simplifying Expressions

Which of the following does not simplify to ?

**Possible Answers:**

All of these simplify to

**Correct answer:**

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x^{2} + 2 = x^{2} + x – 2 – x^{2} + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0

### Example Question #30 : Simplifying Expressions

Simplify the result of the following steps, to be completed in order:

1. Add 7*x* to 3*y*

2. Multiply the sum by 4

3. Add *x* to the product

4. Subtract *x –* *y* from the sum

**Possible Answers:**

28*x* + 13*y*

29*x* + 13*y*

28*x* + 12*y*

28*x* + 11*y*

28*x* – 13*y*

**Correct answer:**

28*x* + 13*y*

Step 1: 7*x* + 3*y*

Step 2: 4 * (7*x* + 3*y*) = 28*x* + 12*y*

Step 3: 28*x* + 12*y* + *x* = 29*x* + 12*y*

Step 4: 29*x* + 12*y* – (*x* – *y*) = 29*x* + 12*y* – *x* + *y* = 28*x* + 13*y*

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