Negative Numbers

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PSAT Math › Negative Numbers

Questions 1 - 10

What is $\frac{-36}{-9}$ ?

Explanation

A negative number divided by a negative number always results in a positive number. divided by equals . Since the answer is positive, the answer cannot be or any other negative number.

What is $\frac{-36}{-9}$ ?

Explanation

A negative number divided by a negative number always results in a positive number. divided by equals . Since the answer is positive, the answer cannot be or any other negative number.

Find the product.

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

Explanation

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

Find the product.

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

Explanation

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

Given that 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 ?

The expression must be negative or zero.

The expression must be positive.

The expression must be negative.

The expression must be positive or zero.

The expression can be positive, negative, or zero.

Explanation

If $-10 \leq a \leq -1$ , then we know that is any number between or equal to and . Therefore must be a negative number.

Also, if $-10 \leq b \leq -1$ , then we know that is any number between or equal to and . Therefore must be a negative number.

Now looking the expression we can find the sign of each component in the expression.

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

Therefore, is a negative number.

Since is between or equal to and we can plug in these end values in to determine the sign of .

Therefore, 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 is negative or zero.

and are positive numbers; is a negative number. All of the following must be positive except:

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

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

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

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

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

Explanation

Since and are positive, all powers of and will be positive; also, in each of the expressions, the powers of and are being added. The clue to look for is the power of and the sign before it.

In the cases of $a^{3}+ b^{4} +c$ and $a+ b^{2} +c^{6}$ , since the negative number is 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 number is 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 number is 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:

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

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

is a positive integer; and are negative integers. Which of the following three expressions must be negative?

I) $b^{a+c}$

II) $b^{a-c}$

III) $b^{c-a}$

None of I, II or III

I, II and III

II and III only

I and III only

I and II only

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 of and , which are not given in the problem.

The correct response is none of these.

Let be a negative integer and be a nonzero integer. Which of the following must be negative regardless of whether is positive or negative?

$ab^{2}$

$a^{2}b$

$a+b^{2}$

$a^{2}+ b$

None of the other answers is correct.

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; if is 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 if is 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.

and are positive numbers; is a negative number. All of the following must be positive except:

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

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

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

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

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

Explanation

Since and are positive, all powers of and will be positive; also, in each of the expressions, the powers of and are being added. The clue to look for is the power of and the sign before it.

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

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

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

is a positive integer; and are negative integers. Which of the following three expressions must be negative?

I) $b^{a+c}$

II) $b^{a-c}$

III) $b^{c-a}$

None of I, II or III

I, II and III

II and III only

I and III only

I and II only

Explanation

The correct response is none of these.

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