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ISEE Upper Level Quantitative : How to add exponential variables

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Variables And Exponents

Simplify:

$5x^{2} y ^{2}+7 + 6x^{2}y^{2} + 11 xy$

Possible Answers:

$11x^{2}y^{2}+ 11 xy +7$

$22x^{3}y^{3}+7$

$29x^{3}y^{3}$

The expression cannot be simplified further

$30x^{2}y^{2}+ 11 xy +7$

Correct answer:

$11x^{2}y^{2}+ 11 xy +7$

Explanation:

Group and combine like terms $5x^{2} y ^{2},6x^{2}y^{2}$ :

$5x^{2} y ^{2}+7 + 6x^{2}y^{2} + 11 xy$

$= 5x^{2} y ^{2}+ 6x^{2}y^{2}+ 11 xy +7$

$=\left ( 5+ 6 \right )x^{2}y^{2}+ 11 xy +7$

$=11x^{2}y^{2}+ 11 xy +7$

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Example Question #2 : Variables And Exponents

x > 0, y < 0

Which is the greater quantity?

(a) $(x + y)^{2}$

(b) $x^{2}+ y^{2}$

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

$(x + y)^{2} = x^{2}+2xy + y^{2}$

Since and have different signs,

xy<0 , and, subsequently,

2xy < 0

Therefore,

$(x + y)^{2} = x^{2}+2xy + y^{2} < x^{2} + y^{2}$

This makes (b) the greater quantity.

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Example Question #3 : Variables And Exponents

Assume that and are not both zero. Which is the greater quantity?

(a) $\frac{(x+y)^{2} + (x - y)^{2}}{x^{2}+y^{2}}$

(b) 4xy

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Simplify the expression in (a):

$\frac{(x+y)^{2} + (x - y)^{2}}{x^{2}+y^{2}}$

$=\frac{(x^{2}+2xy + y^{2})+ (x^{2}-2xy + y^{2})}{x^{2}+y^{2}}$

$=\frac{ x^{2}+x^{2} +2xy -2xy+ y^{2}+ y^{2}}{x^{2}+y^{2}}$

$=\frac{ 2x^{2} +2 y^{2}}{x^{2}+y^{2}} =\frac{ 2(x^{2}+y^{2})}{x^{2}+y^{2}} = 2$

Therefore, whether (a) or (b) is greater depends on the values of and , neither of which are known.

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Example Question #3 : How To Add Exponential Variables

x > 0

Which is the greater quantity?

(a) $x^{2} + 3x$

(b)

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

We give at least one positive value of for which (a) is greater and at least one positive value of for which (b) is greater.

Case 1: x = 2

(a) $x^{2} + 3x = 2^{2} + 3 \cdot 2 = 4 + 6 = 10$

(b) $4x = 4 \cdot 2 = 8$

Case 2: $x = \frac{1}{2}$

(a) $x^{2} + 3x = \left ( \frac{1}{2} \right ) ^{2} + 3 \cdot \frac{1}{2} = \frac{1}{4} + \frac{3}{2} = \frac{7}{4}= 1 \frac{3}{4}$

(b) $4x = 4 \cdot \frac{1}{2} = 2$

Therefore, either (a) or (b) can be greater.

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Example Question #1 : Variables And Exponents

Assume all variables to be nonzero.

Simplify: $\left (12x^{5}y ^{4}z^{3} \right )^{0} + \left (3x^{5}y ^{4}z^{3} \right )^{0}$

Possible Answers:

$15x^{10}y ^{8}z^{6}$

None of the answer choices are correct.

$15x^{5}y ^{4}z^{3}$

$36x^{5}y ^{4}z^{3}$

$36x^{10}y ^{8}z^{6}$

Correct answer:

None of the answer choices are correct.

Explanation:

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

$\left (12x^{5}y ^{4}z^{3} \right )^{0} + \left (3x^{5}y ^{4}z^{3} \right )^{0} = 1 + 1 = 2$ .

None of the given expressions are correct.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to add exponential variables

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Variables And Exponents

Example Question #2 : Variables And Exponents

Example Question #3 : Variables And Exponents

Example Question #3 : How To Add Exponential Variables

Example Question #1 : Variables And Exponents

All ISEE Upper Level Quantitative Resources

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