ISEE Upper Level Quantitative : Variables and Exponents

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

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Example Questions

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Example Question #1 : How To Add Exponential Variables

Simplify:

Possible Answers:

The expression cannot be simplified further

Correct answer:

Explanation:

Group and combine like terms :

Example Question #2 : Variables And Exponents

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

Since  and  have different signs,

, and, subsequently,

Therefore, 

This makes (b) the greater quantity.

Example Question #1 : How To Add Exponential Variables

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

(a) 

(b) 

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Simplify the expression in (a):

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

Example Question #4 : Variables And Exponents

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(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: 

(a) 

(b) 

Case 2: 

(a) 

(b) 

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

Example Question #5 : Variables And Exponents

Assume all variables to be nonzero. 

Simplify: 

Possible Answers:

None of the answer choices are correct.

Correct answer:

None of the answer choices are correct.

Explanation:

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

.

None of the given expressions are correct.

Example Question #6 : Variables And Exponents

Simplify:

Possible Answers:

Correct answer:

Explanation:

Example Question #1 : Variables And Exponents

Which is greater?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(b) is greater

Explanation:

If , then  and 

 

, so by transitivity, , and (b) is greater

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

Expand: 

Which is the greater quantity?

(a) The coefficient of 

(b) The coefficient of 

Possible Answers:

The two quantities are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

The two quantities are equal.

Explanation:

By the Binomial Theorem, if  is expanded, the coefficient of  is

 .

(a) Substitute : The coerfficient of  is 

.

(b) Substitute : The coerfficient of  is 

.

The two are equal.

Example Question #62 : Variables

Which is greater?

(a) 

(b) 

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

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 #63 : Variables

Which is the greater quantity?

(a) 

(b)

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossble to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

Simplify the expression in (a):

Since 

,

making (a) greater.

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