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PSAT Math : How to multiply negative numbers

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Example Questions

Example Question #2 :Negative Numbers

Ifis a positive number, and -3b is also a positive number, what is a possible value for?

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

Because $\dpi{100} \small -3b$ is positive, $\dpi{100} \small b$ 必须be negative since the product of two negative numbers is positive.

Because $\dpi{100} \small ab$ is also positive, $\dpi{100} \small a$ 必须also be negative in order to produce a prositive product.

To check you answer, you can try plugging in any negative number for $\dpi{100} \small a$ .

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Example Question #1 :Negative Numbers

,, andare all negative odd integers. Which of the following three expressions must be positive?

I) $a^{b+c}$

II) $a^{b-c}$

III) $a^{c-b}$

Possible Answers:

II only

I only

All of these

III only

None of these

Correct answer:

All of these

Explanation:

A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.

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Example Question #2 :How To Multiply Negative Numbers

is a positive integer;andare negative integers. Which of the following three expressions必须be negative?

I) $b^{a+c}$

II) $b^{a-c}$

III) $b^{c-a}$

Possible Answers:

I and II only

I, II and III

II and III only

None of I, II or III

I and III only

Correct answer:

None of I, II or III

Explanation:

A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity ofand, which are not given in the problem.

The correct response is none of these.

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Example Question #1 :Negative Numbers

,, andare all negative numbers. Which of the following must be positive?

Possible Answers:

$\frac{ a^{8}c^{3}}{b^{6}}$

$\frac{ b^{5}c^{4}}{a^{12}}$

$\frac{ a^{7}c^{5}}{b^{3}}$

$\frac{a^{4}b^{7}}{c^{5}}$

$\frac{ a^{6}b^{4}}{c^{9}}$

Correct answer:

$\frac{a^{4}b^{7}}{c^{5}}$

Explanation:

The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.

$\frac{ a^{6}b^{4}}{c^{9}}$ : $a^{6}$ and $b^{4}$ are positive, yielding a positive dividend; $c^{9}$ is a negative divisor; this result is negative.

$\frac{ a^{7}c^{5}}{b^{3}}$ : $a^{7}$ and $c^{5}$ are negative, yielding a positive dividend; $b^{3}$ is a negative divisor; this result is negative.

$\frac{ a^{8}c^{3}}{b^{6}}$ : $a^{8}$ is positive and $c^{3}$ is negative, yielding a negative dividend; $b^{3}$ is a positive divisor; this result is negative.

$\frac{ b^{5}c^{4}}{a^{12}}$ : $b^{5}$ is negative and $c^{4}$ is positive, yielding a negative dividend; $a^{12}$ is a positive divisor; this result is negative.

$\frac{a^{4}b^{7}}{c^{5}}$ : $a^{4}$ is positive and $b^{7}$ is negative, yielding a negative dividend; $c^{5}$ is a negative divisor; this result is positive.

The correct choice is $\frac{a^{4}b^{7}}{c^{5}}$ .

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Example Question #4 :How To Multiply Negative Numbers

andare positive numbers;is a negative number. All of the following必须be positive except:

Possible Answers:

$a^{4}+ b^{5} +c^{3}$

$a+ b^{2} +c^{6}$

$a^{3}+ b^{4} +c$

$a^{2} -b^{7} +c$

$a^{2}-b^{3} +c$

Correct answer:

$a^{4}+ b^{5} +c^{3}$

Explanation:

Sinceandare positive, all powers ofandwill be positive; also, in each of the expressions, the powers ofandare being added. The clue to look for is the power ofand the sign before it.

In the cases of $a^{3}+ b^{4} +c$ and $a+ b^{2} +c^{6}$ , since the negative numberis being raised to an even power, each expression amounts to the sum of three positive numbers, which is positive.

In the cases of $a^{2} -b^{7} +c$ and $a^{2}-b^{3} +c$ , since the negative numberis being raised to an odd power, the middle power is negative - but since it is being subtracted, it is the same as if a positive number is being added. Therefore, each is essentially the sum of three positive numbers, which, again, is positive.

In the case of $a^{4}+ b^{5} +c^{3}$ , however, since the negative numberis being raised to an odd power, the middle power is again negative. This time, it is basically the same as subtracting a positive number. As can be seen in this example, it is possible to have this be equal to a negative number:

a = 1, b = -2, c= 1 :

$a^{4}+ b^{5} +c^{3} = 1^{4}+ (-2)^{5} +1^{3} = 1 + (-32)+ 1 = -30$

Therefore, $a^{4}+ b^{5} +c^{3}$ is the correct choice.

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Example Question #5 :How To Multiply Negative Numbers

Letbe a negative integer andbe a nonzero integer. Which of the following必须be negative regardless of whetheris positive or negative?

Possible Answers:

$a^{2}b$

$a^{2}+ b$

$ab^{2}$

$a+b^{2}$

None of the other answers is correct.

Correct answer:

$ab^{2}$

Explanation:

Since $b^{2}$ is positive, $ab^{2}$ , the product of a negative number and a positive number, must be negative also.

Of the others:

$a^{2}b$ is incorrect; ifis negative, then $a^{2}$ is positive, and $a^{2}b$ assumes the sign of.

$a^{2}+ b$ is incorrect; again, $a^{2}$ is positive, and ifis a positive number, $a^{2}+ b$ is positive.

$a+b^{2}$ is incorrect; regardless of the sign of, $b^{2}$ is positive, and if its absolute value is greater than that of, $a+b^{2}$ is positive.

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Example Question #6 :How To Multiply Negative Numbers

Given that a, b are both integers, $-10 \leq a \leq -1$ , and $-10 \leq b \leq -1$ , which of the following is correct about the sign of the expression (a -10)(b +10) ?

Possible Answers:

The expression must be negative.

The expression can be positive, negative, or zero.

The expression must be positive.

The expression must be positive or zero.

The expression must be negative or zero.

Correct answer:

The expression must be negative or zero.

Explanation:

If $-10 \leq a \leq -1$ , then we know thatis any number between or equal toand. Therefore必须be a negative number.

Also, if $-10 \leq b \leq -1$ , then we know thatis any number between or equal toand. Therefore必须be a negative number.

Now looking the expression (a -10)(b +10) we can find the sign of each component in the expression.

Sinceis negative, we know that a negative number minus another number is still a negative number.

Therefore, (a-10) is a negative number.

Sinceis between or equal toandwe can plug in these end values in to determine the sign of (b +10) .

(-10+10)=0

(-1+10)=9

Therefore, (b +10) is either zero or a positive number.

Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that (a -10)(b +10) is negative or zero.

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Example Question #7 :How To Multiply Negative Numbers

Find the product.

$(-270) \times (-131)$

Possible Answers:

35370

35100

Correct answer:

35370

Explanation:

When multiplying together two negatives, our value for the product become positive. $(-270) \times (-131) = 270 \times 131 = 35370$

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Example Question #8 :How To Multiply Negative Numbers

Find the product.

$76 \times -31=$

Possible Answers:

2356

Correct answer:

Explanation:

Since we have one positive and one negative multiple, the resulting product must be negative. $76 \times (-31) = -(76\times 31) = -2356$

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PSAT Math : How to multiply negative numbers

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #2 :Negative Numbers

Example Question #1 :Negative Numbers

Example Question #2 :How To Multiply Negative Numbers

Example Question #1 :Negative Numbers

Example Question #4 :How To Multiply Negative Numbers

Example Question #5 :How To Multiply Negative Numbers

Example Question #6 :How To Multiply Negative Numbers

Example Question #7 :How To Multiply Negative Numbers

Example Question #8 :How To Multiply Negative Numbers

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