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ISEE Upper Level Quantitative : Variables and Exponents

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

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Example Questions

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Example Question #14 : How To Multiply Exponential Variables

and are positive integers greater than 1.

Which is the greater quantity?

(A) $(x+y -1)^{2}$

(B) $(x-y+1)^{2}$

Possible Answers:

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute u = y - 1 . Since y > 1 , must be positive, and this problem is to compare $(x+u)^{2}$ and $(x-u)^{2}$ .

$(x+u)^{2} = x^{2} + 2ux + u^{2}$

and

$(x-u)^{2} = x^{2} - 2ux + u^{2}$

Since 2, , and are positive, by closure, 2ux > 0 > -2ux , and by the addition property of inequality,

$x^{2} + 2ux + u^{2} > x^{2} - 2ux + u^{2}$

$(x+u)^{2} > (x-u)^{2}$

Substituting back:

$(x+y -1)^{2} > (x-y+1)^{2}$

(A) is the greater quantity.

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Example Question #261 : Algebraic Concepts

and are positive integers greater than 1.

Which is the greater quantity?

(A) $\left (x + y + 4\right ) ^{2}$

(B) $\left (x + 2y + 2\right ) ^{2}$

Possible Answers:

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

It is impossible to determine which is greater from the information given

Explanation:

Case 1: x = 2, y = 2

Then

$\left (x + y + 4\right ) ^{2} = \left (2 + 2 + 4\right ) ^{2} = 8 ^{2} = 64$

and

$\left (x + 2y + 2\right ) ^{2} = \left (2 + 2 \cdot 2 + 2\right ) ^{2} = 8 ^{2} = 64$

This makes the quantities equal.

Case 2:

x = 2, y = 3

Then

$\left (x + y + 4\right ) ^{2} = \left (2 + 3 + 4\right ) ^{2} = 9 ^{2} = 81$

and

$\left (x + 2y + 2\right ) ^{2} = \left (2 + 2 \cdot 3 + 2\right ) ^{2} = 10 ^{2} = 100$

This makes (B) greater.

Therefore, it is not clear which quantity, if either, is greater.

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Example Question #262 : Algebraic Concepts

and are positive integers greater than 1.

Which is the greater quantity?

(A) $\left (x + y + 1\right ) ^{2}$

(B) (x+ 1 ) (x + 2y + 1)

Possible Answers:

(B) is greater

It is impossible to determine which is greater from the information given

(A) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute u = x +y+1 . The expressions to be compared are

$\left (x + y + 1\right ) ^{2} = u ^{2}$

and

$(x+ 1 ) (x + 2y + 1) =(u - y ) (u+y)= u^{2}- y^{2}$

Since is positive, so is $y^{2}$ , and

$u^{2}> u^{2}- y^{2}$

Substituting back,

$\left (x + y + 1\right ) ^{2} > (x+ 1 ) (x + 2y + 1)$ ,

making (A) greater.

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Example Question #263 : Algebraic Concepts

Factor:

$4x ^{2} - 9 y^{2} + 14x - 21y$

Possible Answers:

(2x-3y-7)(2x+3y)

(2x+3y-7)(2x+3y)

(2x+3y+7)(2x-3y)

(2x-3y-7)(2x-3y)

The expression is a prime polynomial.

Correct answer:

(2x+3y+7)(2x-3y)

Explanation:

We can rewrite as follows:

$4x ^{2} - 9 y^{2} + 14x - 21y$

$= (4x ^{2} - 9 y^{2}) +( 14x - 21y)$

Each group can be factored - the first as the difference of squares, the second as a pair with a greatest common factor. This becomes

(2x+3y)(2x-3y)+7 ( 2x - 3y) ,

which, by distribution, becomes

(2x+3y+7)(2x-3y)

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Example Question #264 : Algebraic Concepts

is a positive number; is the additive inverse of .

Which is the greater quantity?

(a) $a ^{2} - b^{2}$

(b) $a - b\;$

Possible Answers:

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

If is the additive inverse of , then, by definition,

a+ b = 0 .

$a^{2} - b^{2}$ , as the difference of the squares of two expressions, can be factored as follows:

$a^{2} - b^{2} = (a+b) (a-b )$

Since a+ b = 0 , it follows that

$a^{2} - b^{2} = 0 \cdot (a-b ) = 0$

Another consequence of being the additive inverse of is that

b= -a , so

a - b = a - (-a) = a+ a = 2a

is positive, so is as well.

It follows that $a - b > a ^{2} - b^{2}$ .

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Example Question #1 : How To Divide Exponential Variables

Half of one hundred divided by five and multiplied by one-tenth is __________.

Possible Answers:

Correct answer:

Explanation:

Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.

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Example Question #1 : How To Divide Exponential Variables

Simplify: $\frac{8x^{-6}}{2x^{-3}}$

Possible Answers:

$64x^{3}$

$\frac{x^{3}}{64}$

$4x^{3}$

$\frac{x^{3}}{4}$

$\frac{4}{x^{3}}$

Correct answer:

$\frac{4}{x^{3}}$

Explanation:

$\frac{8x^{-6}}{2x^{-3}} = \frac{8}{2} \cdot \frac{x^{-6}}{x^{-3}} = 4 \cdot x ^{-6 - (-3)} = 4 \cdot x ^{-3} =4 \cdot \frac{1}{x^{3}} = \frac{4}{x^{3}}$

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Example Question #1 : How To Divide Exponential Variables

Simplify:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

Possible Answers:

$\frac{x ^{2} }{2y ^{2} }$

$\frac{x ^{2}y ^{2} }{2 }$

$\frac{1 }{2 x ^{2}y ^{2}}$

$\frac{2}{x ^{2}y ^{2} }$

$\frac{y ^{2} }{2x ^{2} }$

Correct answer:

$\frac{x ^{2} }{2y ^{2} }$

Explanation:

Break the fraction up and apply the quotient of powers rule:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

$= \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$

$= \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$

$= \frac{1 }{2} \cdot x ^{2} \cdot y ^{-2}$

$= \frac{1 }{2} \cdot x ^{2} \cdot \frac{1}{y ^{2}}$

$= \frac{x ^{2} }{2y ^{2} }$

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Example Question #4 : How To Divide Exponential Variables

Simplify: $\frac{16x^4}{12x^2}$

Possible Answers:

$\frac{4x^2}{3}$

16x

$\frac{4}{3x^2}$

$\frac{4}{3}$

$\frac{x^2}{3}$

Correct answer:

$\frac{4x^2}{3}$

Explanation:

To simplify this expression, look at the like terms separately. First, simplify $\frac{16}{12}$ . This becomes $\frac{4}{3}$ . Then, deal with the $\frac{x^4}{x^2}$ . Since the bases are the same and you're dividing, you can subtract exponents. This gives you x^2. Since the exponent is positive, you put in the numerator. This gives you a final answer of $\frac{4x^2}{3}$ .

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Example Question #1 : How To Divide Exponential Variables

is a negative number.

Which is the greater quantity?

(a) The reciprocal of $x^{7}$

(b) The reciprocal of $x^{8}$

Possible Answers:

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

A negative number raised to an odd power is negative; a negative number raised to an even power is positive. It follows that $x^{7}$ is negative and $x^{8}$ is positive. Also, the reciprocal of a nonzero number assumes the same sign as the number itself, so the reciprocal of $x^{8}$ is positive and that of $x^{7}$ is negative. It follows that the reciprocal of $x^{8}$ is the greater of the two.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Variables and Exponents

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #14 : How To Multiply Exponential Variables

Example Question #261 : Algebraic Concepts

Example Question #262 : Algebraic Concepts

Example Question #263 : Algebraic Concepts

Example Question #264 : Algebraic Concepts

Example Question #1 : How To Divide Exponential Variables

Example Question #1 : How To Divide Exponential Variables

Example Question #1 : How To Divide Exponential Variables

Example Question #4 : How To Divide Exponential Variables

Example Question #1 : How To Divide Exponential Variables

All ISEE Upper Level Quantitative Resources

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