Why does a negative times a negative give a positive?

Follow the pattern (-4)(2) = -8, (-4)(1) = -4, (-4)(0) = 0. Every time the second factor decreases by 1, the product increases by 4. Continuing, (-4)(-1) has to be 4. Multiplying by a negative reverses direction, so reversing twice sends you back to the positive side.

Do I ever have to find a common denominator when multiplying or dividing fractions?

No. Common denominators are needed for adding and subtracting. To multiply, go straight across the numerators and denominators; to divide, flip the second fraction and then multiply straight across. Canceling common factors first is optional but keeps the numbers small.

Is -3^2 really different from (-3)^2?

Yes. (-3)^2 = 9 because the parentheses make -3 the base. -3^2 = -9 because the exponent only applies to the 3, and the negative sign is applied to the result. They agree only when the exponent is odd, as with (-2)^3 = -8 and -2^3 = -8.

When I substitute a negative number for a variable, why do I need parentheses?

Because the parentheses tell the exponent (or a nearby multiplication) that the whole negative number is the base. If x = -6, then x^2 means (-6)^2 = 36. Writing -6^2 instead gives -36, a different number. Putting substituted values in parentheses every time prevents that error.

M7MATH-2.3

Multiplying & Dividing Rational Numbers

Learn to multiply and divide positive and negative fractions and decimals: sign rules, flipping to the reciprocal, and why (−3)² and −3² are different.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Multiplying & Dividing Rational Numbers, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You already know how to add and subtract rational numbers. Multiplication and division are, in some ways, easier: instead of tracking which number is "bigger," you multiply or divide the numbers as if they were all positive, then decide on one sign at the end using a single rule about matching signs.

This lesson pulls together four skills that show up constantly for the rest of the year: the same-sign/different-sign rule, multiplying fractions and decimals that carry negative signs, turning every division problem into a multiplication problem with a reciprocal, and reading exponent notation carefully enough to tell

(-3)^2

apart from

-3^2

. That last one looks like a trick, but it is really about what the exponent is attached to — and it is where the most homework mistakes happen.

The Same-Sign, Different-Sign Rule

For multiplication and division, ignore the signs first and work with the sizes of the numbers. Then apply one rule: if the two signs are the same, the answer is positive; if the signs are different, the answer is negative.

Expression	Signs	Result
$(-6)(-4)$	same	$24$
$(-6)(4)$	different	$-24$
$-56 \div (-8)$	same	$7$
$56 \div (-8)$	different	$-7$

Why should two negatives make a positive? Think of

(-4)(3) = -12

(-4)(2) = -8

(-4)(1) = -4

(-4)(0) = 0

. Each time the second factor drops by 1, the product rises by 4. Continuing the pattern,

(-4)(-1)

must be

4

and

(-4)(-2)

must be

8

. The positive result is what keeps arithmetic consistent.

When more than two factors appear, count the negative factors. An even number of negative factors gives a positive product; an odd number gives a negative product. So

(-2)(-3)(-1)

has three negatives and equals

-6

, while

(-2)(-3)(-1)(-1)

has four and equals

6

.

A common mistake is importing the addition rules here. With addition,

-5 + (-3) = -8

stays negative. With multiplication,

(-5)(-3) = 15

becomes positive. The operations follow different rules, so name the operation before you decide the sign.

One more fact: zero times anything is

0

, and

0 \div (-7) = 0

. But

-7 \div 0

is undefined — you can never divide by zero.

Multiplying Fractions and Decimals

To multiply fractions, multiply numerator times numerator and denominator times denominator, then simplify. Handle the sign separately.

\left(-\frac{4}{9}\right)\left(-\frac{3}{8}\right) = +\frac{4 \cdot 3}{9 \cdot 8} = \frac{12}{72} = \frac{1}{6}

Both signs match, so the product is positive. You can also cancel common factors before multiplying: the

4

and

8

share a factor of

4

, and the

3

and

9

share a factor of

3

, leaving

\frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}

. Canceling first keeps the numbers small.

Mixed numbers must become improper fractions first.

-2\frac{1}{2}

-\frac{5}{2}

, not

-2 + \frac{1}{2}

in disguise — the negative applies to the whole quantity

2\frac{1}{2}

\left(-2\frac{1}{2}\right)\left(\frac{4}{15}\right) = -\frac{5}{2}\cdot\frac{4}{15} = -\frac{20}{30} = -\frac{2}{3}

For decimals, multiply as with whole numbers and count decimal places. In

(-0.6)(0.4)

, the digits give

6 \cdot 4 = 24

, and there are two decimal places total, so the size is

0.24

. The signs differ, so the answer is

-0.24

.

Where students go wrong: losing the negative sign partway through a multi-step problem. A good habit is to write the sign of the answer down first, off to the side, then compute the size. Notice also that multiplying by a number between

0

and

1

makes the size smaller —

0.24

is closer to zero than

0.6

is — which is a quick way to check that your answer is reasonable.

Dividing by Multiplying by the Reciprocal

Every division problem can be rewritten as multiplication. The reciprocal (or multiplicative inverse) of a nonzero number is the number you multiply it by to get

1

. The reciprocal of

\frac{3}{5}

\frac{5}{3}

; the reciprocal of

-\frac{2}{7}

-\frac{7}{2}

; the reciprocal of

-4

, which is

-\frac{4}{1}

, is

-\frac{1}{4}

.

To divide, keep the first number, change

\div

\times

, and flip the second number:

-\frac{3}{4} \div \left(-\frac{9}{10}\right) = -\frac{3}{4} \cdot \left(-\frac{10}{9}\right) = +\frac{30}{36} = \frac{5}{6}

Notice that flipping does not change the sign. The reciprocal of a negative number is still negative, so you apply the same-sign rule after flipping, exactly as with any multiplication.

A fraction bar means division, so signs can be written in three equivalent places:

\frac{-8}{5} = \frac{8}{-5} = -\frac{8}{5}

All three name the same number. That flexibility is useful later when you simplify expressions.

For decimal division, one clean approach is to convert to fractions, but you can also use the standard method:

-1.44 \div 0.12

becomes

-144 \div 12 = -12

after multiplying both numbers by

100

. Multiplying dividend and divisor by the same power of ten does not change the quotient.

Two frequent errors: flipping the first fraction instead of the second, and flipping while still dividing. Say the steps out loud — keep, change, flip — and check that the sign was decided from the two original signs.

Reading Exponents: (−3)² Versus −3²

An exponent applies only to whatever is immediately to its left. In

(-3)^2

, the parentheses make

-3

the base, so the expression means

(-3)(-3) = 9

. In

-3^2

, the base is just

3

, and the negative sign stays outside: it means

-(3 \cdot 3) = -9

. You can read

-3^2

as "the opposite of

3

squared."

Expression	Base	Meaning	Value
$(-3)^2$	$-3$	$(-3)(-3)$	$9$
$-3^2$	$3$	$-(3 \cdot 3)$	$-9$
$(-3)^3$	$-3$	$(-3)(-3)(-3)$	$-27$
$-3^3$	$3$	$-(3 \cdot 3 \cdot 3)$	$-27$
$(-2)^4$	$-2$	four factors of $-2$	$16$

The last two rows of the first three show something worth noticing: with an odd exponent,

(-3)^3

and

-3^3

happen to agree, because an odd number of negative factors is negative anyway. With an even exponent they never agree, and that is where confusion shows up on homework.

So for a negative base in parentheses, use the factor-counting idea: even exponent gives a positive value, odd exponent gives a negative value.

(-1)^{10} = 1

and

(-1)^{11} = -1

.

When you substitute a negative value into an expression, always add parentheses. If

x = -5

and you need

x^2

, write

(-5)^2 = 25

. Typing

-5^2

gives

-25

and changes the answer. This habit matters again in the next lesson on order of operations, where exponents are evaluated before multiplication and division.

Key terms

Rational number.: Any number that can be written as a ratio $\frac{a}{b}$ of two integers with $b \neq 0$ . This includes integers, fractions, mixed numbers, and terminating or repeating decimals, positive or negative.
Reciprocal (multiplicative inverse).: The number that multiplies with a given nonzero number to give $1$ . The reciprocal of $-\frac{5}{8}$ is $-\frac{8}{5}$ . Zero has no reciprocal.
Same-sign rule.: When two factors or a dividend and divisor have the same sign, the product or quotient is positive.
Different-sign rule.: When two factors or a dividend and divisor have opposite signs, the product or quotient is negative.
Base.: The number a power is built from. In $(-3)^2$ the base is $-3$ ; in $-3^2$ the base is $3$ and the negative sign is applied afterward.
Quotient.: The result of a division. Because $\frac{a}{b}$ means $a \div b$ , a fraction bar is a division symbol.
Undefined.: A description of an expression with no numerical value, such as any division by zero. Note that $0 \div 5 = 0$ , but $5 \div 0$ is undefined.
Opposite (additive inverse).: The number the same distance from zero on the other side, such as $7$ and $-7$ . Writing a negative sign in front of an expression means "take the opposite of it."

Worked example

Evaluate

-3^2 \cdot \left(-\frac{2}{3}\right) \div (-0.5)

Start with the power. The exponent in

-3^2

is attached to the

3

only, so this means

-(3 \cdot 3) = -9

. The expression is now

-9 \cdot \left(-\frac{2}{3}\right) \div (-0.5)

.

Next, multiply. Decide the sign first:

-9

and

-\frac{2}{3}

have the same sign, so the product is positive. Now the sizes:

9 \cdot \frac{2}{3} = \frac{18}{3} = 6

. So the product is

+6

, and the expression becomes

6 \div (-0.5)

.

Finally, divide. Rewrite

-0.5

as the fraction

-\frac{1}{2}

, whose reciprocal is

-2

. Keep, change, flip:

6 \div \left(-\frac{1}{2}\right) = 6 \cdot (-2)

. The signs differ, so the answer is negative, and

6 \cdot 2 = 12

.

The value is

-12

.

Sanity check: dividing by a number between

-1

and

0

should produce an answer bigger in size than

6

and negative, and

-12

fits. A common wrong answer here is

12

, which comes from treating

-3^2

9

Practice questions

Which expression has a value of

-16

$(-4)^2$
$-4^2$
$(-2)^4$
$(-4)(-4)$

Answer: $-4^2$

-4^2

the exponent belongs to the

4

alone, so the expression means

-(4 \cdot 4) = -16

. The other three all equal

16

(-4)^2

and

(-4)(-4)

are two negative factors multiplied, and

(-2)^4

has four negative factors, an even number, so it is positive. The lesson here is that parentheses, not the printed order of symbols, decide the base.

Simplify

-\frac{7}{8} \div \left(-\frac{7}{12}\right)

and show the reciprocal step.

Answer: $\frac{3}{2}$ (or $1\frac{1}{2}$ )

Keep the first number, change division to multiplication, and flip the second:

-\frac{7}{8} \cdot \left(-\frac{12}{7}\right)

. Both signs are negative, so the answer is positive. Cancel the common factor of

7

, and cancel

12

with

8

using the factor

4

\frac{1}{2} \cdot \frac{3}{1} = \frac{3}{2}

. A useful check: dividing a number by something smaller in size than itself gives a quotient bigger than

1

, and

\frac{3}{2}

is.

Between 8 p.m. and 10:30 p.m., the temperature outside changed by

-7.5

degrees Celsius. Write a division expression for the average change per hour, evaluate it, and explain what the sign of your answer means.

Answer: $-7.5 \div 2.5 = -3$ , so the temperature fell an average of 3 degrees Celsius each hour.

The elapsed time from 8:00 to 10:30 is

2.5

hours, and average rate of change is total change divided by time, so the expression is

-7.5 \div 2.5

. Multiply both numbers by

10

to clear decimals:

-75 \div 25 = -3

. The signs are different (negative divided by positive), which confirms a negative quotient. A negative rate means the quantity is decreasing, so the temperature dropped 3 degrees per hour on average. Writing the units in the answer, degrees per hour, is part of a complete response.

FAQ

Why does a negative times a negative give a positive?: Follow the pattern $(-4)(2) = -8$ , $(-4)(1) = -4$ , $(-4)(0) = 0$ . Every time the second factor decreases by $1$ , the product increases by $4$ . Continuing, $(-4)(-1)$ has to be $4$ . Multiplying by a negative reverses direction, so reversing twice sends you back to the positive side.
Do I ever have to find a common denominator when multiplying or dividing fractions?: No. Common denominators are needed for adding and subtracting. To multiply, go straight across the numerators and denominators; to divide, flip the second fraction and then multiply straight across. Canceling common factors first is optional but keeps the numbers small.
Is $-3^2$ really different from $(-3)^2$ ?: Yes. $(-3)^2 = 9$ because the parentheses make $-3$ the base. $-3^2 = -9$ because the exponent only applies to the $3$ , and the negative sign is applied to the result. They agree only when the exponent is odd, as with $(-2)^3 = -8$ and $-2^3 = -8$ .
When I substitute a negative number for a variable, why do I need parentheses?: Because the parentheses tell the exponent (or a nearby multiplication) that the whole negative number is the base. If $x = -6$ , then $x^2$ means $(-6)^2 = 36$ . Writing $-6^2$ instead gives $-36$ , a different number. Putting substituted values in parentheses every time prevents that error.

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