Algebra - GMAT Quantitative

Card 0 of 3752

Question

$0.04 ^{t} = 25 ^{x+2}$

Express in terms of .

Answer

$0.04 ^{t} = 25 ^{x+2}$

$\left ( \frac{1}{25} \right )^{t} = 25 ^{x+2}$

$\left ( 25 ^{-1} \right )^{t} = 25 ^{x+2}$

$25 ^{- t} = 25 ^{x+2}$

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Question

Simplify: $\frac{x^6y^{-4}z^0}{x^{-5}y^{10}z^{-6}}$

Answer

To solve, we must first simplify the negative exponents by shifting them to the other side of the fraction:

$\frac{x^6y^{-4}z^0}{x^{-5}y^{10}z^{-6}} = \frac{x^6x^5z^0z^6}{y^{10}y^4}$

Then we can simply the multiplied like bases by adding their exponents:

$\frac{x^6x^5z^0z^6}{y^{10}y^4} = \frac{x^{11}z^6}{y^{14}}$

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Question

$\frac{(2^6 \cdot 2^7)^3}{2^{40}} =$

Answer

First we can simplify the numerator's parentheses by adding the like bases' exponents:

$\frac{(2^6 \cdot 2^7)^3}{2^{40}} = \frac{(2^{13})^3}{2^{40}}$

We can then simplify the numerator further by multiplying the base's exponent by the exponent to which it is raised:

$\frac{(2^{13})^3}{2^{40}} = \frac{2^{39}}{2^{40}}$

We can then subtract the denominator's exponent from the numerator's:

$\frac{1}{2^1} = \frac{1}{2}$

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Question

Simplify the following expression:

$\frac{343y^7}{49y^4}-9y^3$

Answer

Simplify the following expression:

$\frac{343y^7}{49y^4}-9y^3$

Let's begin by simplifying the fraction. Recall that when dividing exponents of similar base, we need to subtract the exponents. We can treat the 343 and the 49 just like regular fractions.

$\frac{343y^7}{49y^4}-9y^3=7y^3-9y^3=-2y^3$

Note that then we perform the subtraction step to get our final answer:

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Question

is the additive inverse of ; is the multiplicative inverse of . Which of the following is equal to the expression

$(A+B)^{C+D}$

regardless of the values of the variables?

Answer

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

, or $D = \frac{1}{C}$ .

It follows that

$(A+B)^{C+D} = 0^{C+ \frac{1}{C}}$ .

0 raised to any nonzero power is equal to 0, and $C + \frac{1}{C}$ must be nonzero, so

$0^{C+ \frac{1}{C}} = 0$ , the correct response.

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Question

is the additive inverse of ; is the additive inverse of . Which of the following is equal to the expression

$(A+B)^{C+D}$

regardless of the values of the variables?

Answer

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of and is the additive inverse of ,

and

$(A+B)^{C+D} = 0^{0}$ ,

which is an undefined expression.

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Question

is the multiplicative inverse of ; is the additive inverse of . Which of the following is equal to the expression

$(A+B)^{C+D}$

regardless of the values of the variables?

Answer

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

, or $B = \frac{1}{A}$ .

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

It follows that

$(A+B)^{C+D} = (A+\frac{1}{A})^{0}$

Any nonzero number raised to the power of 0 is equal to 1. Therefore,

$(A+\frac{1}{A})^{0} = 1$ , the correct choice.

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Question

is the additive inverse of ; is the multiplicative inverse of . Which of the following is equal to the expression

$(AB)^{CD}$

regardless of the values of the variables?

Answer

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

, or

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

.

It follows that

$[A \cdot (-A)] ^{1} =( -A ^{2} )^{1} = -A ^{2}$ , the correct response.

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Question

Find the domain of $y=\sqrt{x^{2}-4}$ .

Answer

We want to see what values of x satisfy the equation. $x^{2}-4$ is under a radical, so it must be positive.

$x^{2}-4\geq 0$

$x^{2}\geq 4$

$x\geq 2, x\leq -2$

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Question

is the multiplicative inverse of ; is the additive inverse of . Which of the following is equal to the expression

$(AB)^{CD}$

regardless of the values of the variables?

Answer

The multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1. Since is the multiplicative inverse of , then

.

The additive inverse of a number is the number which, when added to that number, yields sum 0. Since is the additive inverse of ,

, or .

It follows that

$(AB)^{CD} = 1^{C(-C)}= 1^{-C^{2}} = 1$ , the correct response.

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Question

is the additive inverse of the multiplicative inverse of ; is the additive inverse of the multiplicative inverse of . Which of the following is equal to the expression

$(AB)^{CD}$

regardless of the values of the variables?

Answer

The additive inverse of a number is the number which, when added to that number, yields sum 0; the multiplicative inverse of a number is the number which, when multiplied by that number, yields product 1.

Let be the multiplicative inverse of . Then

, or, equivalently, $X = \frac{1}{B}$ .

is the additive inverse of this number, so

$A + \frac{1}{B}= 0$

$A = - \frac{1}{B}$

$A \cdot B = - \frac{1}{B} \cdot B$

By similar reasoning, , and

$(AB)^{CD} = \left ( -1\right ) ^{-1} = \frac{1}{-1} = -1$

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Question

$FG= 2^{100}$

$HJ = 2^{10}$

$\frac{F}{J} = 4^{40}$

Which of the following is equal to $\frac{H}{G}$ ?

Answer

Divide:

$\frac{HJ}{FG}=\frac{ 2^{10}}{2^{100}} = 2^{10-100} = 2^{-90}$

$\frac{HJ}{FG}= \frac{H}{G} \cdot \frac{J}{F} = \frac{H}{G} \div \frac{F} {J}$

$\frac{F}{J} = 4^{40} = (2 ^{2})^{40} = 2 ^{2 \cdot 40 } = 2 ^{80}$

Substitute:

$\frac{H}{G} \div \frac{F} {J} = \frac{HJ}{FG}$

$\frac{H}{G} \div 2 ^{80}= 2^{-90}$

$\frac{H}{G} \div 2 ^{80}\cdot 2 ^{80} = 2^{-90} \cdot 2 ^{80}$

$\frac{H}{G} = 2^{-90+ 80} = 2^{-10} = \frac{1}{2^{10}}$

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Question

How many integers $\dpi{100} \small (x)$ can complete this inequality?

$7< 2x-3 <15$

Answer

$7< 2x-3 <15$

3 is added to each side to isolate the $\dpi{100} \small x$ term:

$10< 2x <18$

Then each side is divided by 2 to find the range of $\dpi{100} \small x$ :

$5< x <9$

The only integers that are between 5 and 9 are 6, 7, and 8.

The answer is 3 integers.

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Question

Solve the inequality:

$-3x+7\geq13$

Answer

$-3x+7\geq13$

$-3x\geq6$

When multiplying or dividing by a negative number on both sides of an inequality, the direction of the inequality changes.

$x\leq-2$

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Question

Solve $5 < 3x + 10 \leq 16$ .

Answer

$5 < 3x + 10 \leq 16$

Subtract 10: $-5 < 3x \leq 6$

Divide by 3: $-5/3 < x \leq 2$

We must carefully check the endpoints. is greater than $-\frac{5}{3}$ and cannot equal $-\frac{5}{3}$ , yet CAN equal 2. Therefore $-\frac{5}{3}$ should have a parentheses around it, and 2 should have a bracket: is in

$\left ( -\frac{5}{3}, 2 \right ]$

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Question

Solve .

Answer

Subtract 3 from both sides:

Divide both sides by :

Remember: when dividing by a negative number, reverse the inequality sign!

Now we need to decide if our numbers should have parentheses or brackets. is strictly greater than , so should have a parentheses around it. Since there is no upper limit here, is in $(-2, \infty )$ .

Note: Infinity should ALWAYS have a parentheses around it, NEVER a bracket.

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Question

Solve $(x-1)^{2}(x+4)<0$ .

Answer

$(x-1)^{2}$ must be positive, except when . When , $(x-1)^{2}=0$ .

Then we know that the inequality is only satisfied when , and $x\neq 1$ . Therefore , which in interval notation is $(-\infty , -4)$ .

Note: Infinity must always have parentheses, not brackets. has a parentheses around it instead of a bracket because is less than , not less than or equal to .

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Question

Solve .

Answer

The roots we need to look at are $x=0, x=1,\ and\ x=-1.$

:

Try

, so

does not satisfy the inequality.

:

Try $x=-\frac{1}{2}$

$-\frac{1}{2}\cdot (-\frac{1}{2}-1)(-\frac{1}{2}+1)=\frac{3}{8}>0$

so does satisfy the inequality.

:

Try $x=\frac{1}{2}$

$\frac{1}{2}\left (\frac{1}{2}-1 \right )\left ( \frac{1}{2}+1 \right )=-\frac{3}{8}<0$

so does not satisfy the inequality.

:

Try .

so satisfies the inequality.

Therefore the answer is and .

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Question

Find the solution set for :

Answer

Subtract 7:

Divide by -1. Don't forget to switch the direction of the inequality signs since we're dividing by a negative number:

$-3 \cdot (-1) > -x \cdot (-1) > 8\cdot (-1)$

Simplify:

or in interval form, .

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Question

Which of the following is equivalent to ?

Answer

To solve this problem we need to isolate our variable .

We do this by subtracting from both sides and subtracting from both sides as follows:

Now by dividing by 3 we get our solution.

or

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