ISEE Upper Level Quantitative : Variables and Exponents

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

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Example Questions

Example Question #211 : Algebraic Concepts

Simplify:

\(\displaystyle 5x^{2} y ^{2}+7 + 6x^{2}y^{2} + 11 xy\)

Possible Answers:

The expression cannot be simplified further

\(\displaystyle 11x^{2}y^{2}+ 11 xy +7\)

\(\displaystyle 22x^{3}y^{3}+7\)

\(\displaystyle 29x^{3}y^{3}\)

\(\displaystyle 30x^{2}y^{2}+ 11 xy +7\)

Correct answer:

\(\displaystyle 11x^{2}y^{2}+ 11 xy +7\)

Explanation:

Group and combine like terms \(\displaystyle 5x^{2} y ^{2},6x^{2}y^{2}\):

\(\displaystyle 5x^{2} y ^{2}+7 + 6x^{2}y^{2} + 11 xy\)

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

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

\(\displaystyle =11x^{2}y^{2}+ 11 xy +7\)

Example Question #51 : Variables

\(\displaystyle x > 0, y < 0\)

Which is the greater quantity?

(a) \(\displaystyle (x + y)^{2}\)

(b) \(\displaystyle x^{2}+ y^{2}\)

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

\(\displaystyle (x + y)^{2} = x^{2}+2xy + y^{2}\)

Since \(\displaystyle x\) and \(\displaystyle y\) have different signs,

\(\displaystyle xy< 0\), and, subsequently,

\(\displaystyle 2xy < 0\)

Therefore, 

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

This makes (b) the greater quantity.

Example Question #1 : How To Add Exponential Variables

Assume that \(\displaystyle x\) and \(\displaystyle y\) are not both zero. Which is the greater quantity?

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

(b) \(\displaystyle 4xy\)

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):

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

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

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

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

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

Example Question #1 : Variables And Exponents

\(\displaystyle x > 0\)

Which is the greater quantity?

(a) \(\displaystyle x^{2} + 3x\)

(b) \(\displaystyle 4x\)

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

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

Case 1: \(\displaystyle x = 2\)

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

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

Case 2: \(\displaystyle x = \frac{1}{2}\)

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

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

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

Example Question #1 : Variables And Exponents

Assume all variables to be nonzero. 

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

Possible Answers:

\(\displaystyle 15x^{10}y ^{8}z^{6}\)

None of the answer choices are correct.

\(\displaystyle 36x^{10}y ^{8}z^{6}\)

\(\displaystyle 36x^{5}y ^{4}z^{3}\)

\(\displaystyle 15x^{5}y ^{4}z^{3}\)

Correct answer:

None of the answer choices are correct.

Explanation:

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

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

None of the given expressions are correct.

Example Question #1 : How To Find The Exponent Of Variables

Simplify:

\(\displaystyle \left (\frac{x^{3}}{2y^{4}} \right ) ^{-3}\)

Possible Answers:

\(\displaystyle \frac{8x^{6} }{ y^{7}}\)

\(\displaystyle \frac{8x^{9} }{ y^{12}}\)

\(\displaystyle \frac{1}{2y}\)

\(\displaystyle \frac{8 y^{7} }{x^{6} }\)

\(\displaystyle \frac{8 y^{12} }{x^{9} }\)

Correct answer:

\(\displaystyle \frac{8 y^{12} }{x^{9} }\)

Explanation:

\(\displaystyle \left (\frac{x^{3}}{2y^{4}} \right ) ^{-3} = \left (\frac{2y^{4}}{x^{3}} \right ) ^{3} = \frac{2^{3} \left (y^{4}\right ) ^{3} }{\left (x^{3} \right )^{3}}= \frac{8 y^{4 \cdot 3} }{x^{3\cdot 3} }= \frac{8 y^{12} }{x^{9} }\)

Example Question #2 : How To Find The Exponent Of Variables

\(\displaystyle y>1\)

Which is greater?

(a) \(\displaystyle y ^{-2}\)

(b) \(\displaystyle y^{2}\)

Possible Answers:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

If \(\displaystyle y > 1\), then \(\displaystyle y ^{2}> 1^{2} = 1\) and \(\displaystyle y ^{-2}= \frac{1}{y ^{2}} < \frac{1}{1} = 1\)

 

\(\displaystyle y ^{2}> 1> y ^{-2}\), so by transitivity, \(\displaystyle y ^{2}> y ^{-2}\), and (b) is greater

Example Question #61 : Variables

Expand: \(\displaystyle (x + 1)^{10}\)

Which is the greater quantity?

(a) The coefficient of \(\displaystyle x^{3}\)

(b) The coefficient of \(\displaystyle x^{7}\)

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

The two quantities are equal.

Correct answer:

The two quantities are equal.

Explanation:

By the Binomial Theorem, if \(\displaystyle (x + 1)^{n}\) is expanded, the coefficient of \(\displaystyle x^{r}\) is

 \(\displaystyle _{n}\textrm{C} _{r} = \frac{n!}{(n-r)!r!}\).

(a) Substitute \(\displaystyle n = 10, r = 3\): The coerfficient of \(\displaystyle x^{3}\) is 

\(\displaystyle _{10}\textrm{C} _{3} = \frac{10!}{(10-3)!3!} = \frac{10!}{7!3!}\).

(b) Substitute \(\displaystyle n = 10, r = 7\): The coerfficient of \(\displaystyle x^{7}\) is 

\(\displaystyle _{10}\textrm{C} _{7} = \frac{10!}{(10-7)!7!} = \frac{10!}{3!7!} = \frac{10!}{7!3!}\).

The two are equal.

Example Question #221 : Algebraic Concepts

Which is greater?

(a) \(\displaystyle \left (-2 \right )^{-3}\)

(b) \(\displaystyle \left ( \frac{1}{2} \right ) ^{3}\)

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

\(\displaystyle \left (-2 \right )^{-3} = \left ( -\frac{1}{2} \right )^{3}\)

A negative number to an odd power is negative, so the expression in (a) is negative. The expression in (b) is positive since the base is positive. (b) is greater.

Example Question #3 : Variables And Exponents

\(\displaystyle x>0\)

Which is the greater quantity?

(a) \(\displaystyle (x + 5)^{3}- (x - 5)^{3}\)

(b) \(\displaystyle 250\)

Possible Answers:

(b) is greater.

It is impossble to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

Simplify the expression in (a):

\(\displaystyle (x + 5)^{3}- (x - 5)^{3}\)

\(\displaystyle =(x^{3}+ 3 \cdot x^{2} \cdot 5 + 3 \cdot x \cdot 5^{2}+ 5^{3}) -(x^{3}- 3 \cdotx^{2} \cdot 5 + 3 \cdot x \cdot 5^{2}- 5^{3})\)

\(\displaystyle =(x^{3}+15 x^{2}+75 x + 125) - (x^{3}-15 x^{2}+75 x- 125)\)

\(\displaystyle =x^{3}-x^{3}+15 x^{2}+15 x^{2}+75 x-75 x + 125+125\)

\(\displaystyle =30 x^{2} + 250\)

Since \(\displaystyle x>0\)

\(\displaystyle (x + 5)^{3}- (x - 5)^{3}=30 x^{2} + 250 > 250\),

making (a) greater.

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