**Answers:**

1. What is the product of −4 and −8?

**Answer: 32** Remember that the product of two negative
numbers is positive, and the product

of a negative and a positive is negative.

2. What is 28 ÷ (−7)?

**Answer: −4 **The properties for dividing positive and
negative numbers is the same as for

multiplication.

3. What is

**Answer:** Adding and
subtracting fractions with the same denominator is easy. Here,

we have 17 fifteenths and we’re subtracting 8 fifteenths, which leaves 9
fifteenths. The Math

Placement Exam is a multiple choice test, and the correct answer may not be in
reduced

form.

4. What is

**Answer:**Multiplying and
dividing fractions is also easy. To multiply, you multiply the

numerators and you multiply the denominators. In this case, we are dividing by
6/11, which

is equivalent to multiplying by 11/6. In other words, we invert and multiply.

5. Simplify

**Answer: 3 **Remember to use the standard order of operations
(grouping symbols, then

exponents and radicals , then multiplication/division, and finally addition and
multiplication.

In this case, the division bar tells us to compute the numerator and denominator
before

dividing. In the numerator, we do the multiplication first, to get 19 − 10,
which simplifies

to 9. Finally, 9 divided by 3 equals 3.

6. What is

**Answer: 29/30** To add these fractions, we must first
convert both fractions so that they have

the same denominator, which in this case can be 30. This gives us
This is now

similar to problem 3, and we get 29/30.

7. Round 177.546811 to three decimal places (that is,
round to three digits

to the right of the decimal point.)

**Answer: 177.547** To round correctly, we look at the next
digit. If the next digit is 5 or more,

we round up. Otherwise, we round down. In this case, the next digit is 8, so we
round the

6 up to 7.

8. Convert 0.3547 to a fraction.

**Answer: 3547/10000** Remember that the digits to the right
of the decimal point represent tenths,

hundredths, thousandths, etc. In this case, the fourth digit represents
ten-thousandths, and

we have 3,547 of these ten-thousandths.

9. Evaluate the expression x^{2} − 5x + 3 for x = −1.

**Answer: 9** It’s a good idea to always substitute into
parentheses. Here, we get (−1)^{2} −

5(−1) + 3 = 1 + 5 + 3 = 9.

10. Write the following without parentheses −3(x^{2} − 3x +
4).

**Answer: −3x**^{2}+9x−12 We’re distributing here, and the −3
must be multiplied times every

term inside the parentheses.

11. Multiply out (x − 3)(x + 5).

**Answer: x**^{2} + 2x − 15 Some would call this FOIL-ing. In
general, when multiplying poly-

nomials by polynomials, we multiply every term in the first factor times every
term in the

second. Here, we get x · x + x · 5 + (−3) · x + (−3) · 5 = x^{2} + 5x − 3x − 15 =
x^{2} + 2x − 15.

12. Multiply out (x − 5)^{2}.

**Answer: x**^{2} − 10x + 25 The result of this is a perfect
square trinomial, which we know will

take a special form. Alternatively, we can simply multiply (x−5)(x−5) as we did
in problem

11.

13. Simplify x^{4} · x^{3}.

**Answer: x**^{7} Here, we have 4 factors of x times 3 factors of x, which is 7
factors altogether.

Symbolically, when we multiply exponential expressions with the same bases, we
add the

exponents.

14. Simplify (x^{3})^{5}.

**Answer: x**^{15} Here, we have 3 factors of x multiplied 5 times, which is 15
factors altogether.

Symbolically, when we raise an exponential expression to another exponent, we
multiply

exponents.

15. Simplify

**Answer: x**^{5} Here, we have 9 factors of x, and we are
dividing by 4 factors of x. This leaves

5 factors. Symbolically, when we divide exponential expressions with the same
bases, we

subtract exponents.

16. Simplify (−1)^{4}.

**Answer: 1 **Remember that when multiplying an even number of
negative numbers together,

the result is positive. If there are an odd number of negatives, the result is
negative.

17. Solve the equation 3x + 4 = 10.

**Answer: x = 2** When solving linear equations, we do the
same thing to both sides to try to

get x by itself on one side. In this case, we can subtract 4 from both sides to
get 3x = 6.

Then we divide both sides by 3 to get x = 2. At this point it is obvious that x
must be 2.

18. What is 50% of 70?

**Answer: 35** A percent, like 50% can be thought of as a
fraction, 50/100 = 0.50. To find “a

percentage of ..., ” we multiply. In this case, we have 0.50 · 70 = 35. On the
Placement

Exam, you’ll need to be able to do this without a calculator .

19. Of x = −2, x = −1, x = 0, x = 1, and x = 3, which are
solutions to

the inequality 3x^{2} ≥ 12?

**Answer: x = −2, 3** There are infinitely many solutions to
this inequality, but this question

is only asking which of these numbers are solutions. Plugging x = −2 into the
left side, we

get 3(−2)^{2} = 12, which is equal to, and therefore greater than or equal to, 12,
so x = −2 is

a solution. For x = −1, the left side is 3(−1)^{2} = 3, which is not greater than
or equal to 12,

so x = −1 is not a solution. Similarly, you will find that x = 0 and x = 1 are
not solutions,

and x = 3 is a solution.

20. Simplify | 7 − 11 |.

**Answer: 4** You can think of an absolute value as being a
distance from zero, and as a result

must be zero or positive . In this case, we have | 7 − 11 | = | − 4 | = 4.

21. Graph the equation −2x + 5y = 10.

**Answer: a.**

To graph a line, you need to find at least two points on the line (i.e., two
solutions to the

equation). Two easy points take the form (0, y) and (x, 0). For (0, y), if we
substitute into

the equation, we get −2(0) + 5y = 10, which becomes 5y = 10 and y = 2. The point
(0, 2),

therefore, is a point on the line. Similarly, you can find that (−5, 0) is also
a point on the

line. Plotting these two points and drawing a straight line through them, gives
you the graph

shown above.

22. Graph the equation

**Answer: a.**

This equation can be graphed as in problem 21, but this equation is in
“slope-intercept”

form, y = mx+b. In this case, we know that b is the y-intercept, so (0, b) = (0,
2) is a point

on the line. Since the slope is m = 2/5 , we can count from the y-intercept “up
2, right 5” to

find another point, (5, 4). Drawing the line through these points gives you the
graph. The

line is the same as the one from problem 21, but on the Placement Exam, this
won’t be the

case.

23 What is the slope of the line pictured?

**Answer: m = 5/3** The slope is the change in y over the
change in x between any pair of points

on the line. Looking at the graph, the easiest point to see are (0, 5) and (−3,
0). The change

in y is 0−5 = −5, and the change in x is −3−0 = −3. The slope, therefore, is

24. A distance of 1 inch represents 15 miles. If two
points on the ground

are 45 miles apart, how far are they apart on the map?

**Answer: 3 inches **Let’s say the distance we’re looking for
is x inches. The distances on the

map are proportional to the actual distances so x is to 45 as 1 is to 15. In
other words,

Solving this equation yields x = 3.

25. Substitute m = −4, n = −2, and s = 3 into the
expression

**Answer: 3** Here, we simply substitute the values into the
expression and simplify. Be sure

to follow the order of operations .