ISEE Upper Level Quantitative : How to find the exponent of variables

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 Find The Exponent Of Variables

Simplify:

Possible Answers:

Correct answer:

Explanation:

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

Which is greater?

(a) 

(b) 

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

If , then  and 

 

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

Example Question #3 : How To Find The Exponent Of Variables

Expand: 

Which is the greater quantity?

(a) The coefficient of 

(b) The coefficient of 

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

The two quantities are equal.

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 #2 : How To Find The Exponent Of Variables

Which is greater?

(a) 

(b) 

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

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 #3 : How To Find The Exponent Of Variables

Which is the greater quantity?

(a) 

(b)

Possible Answers:

It is impossble to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

Simplify the expression in (a):

Since 

,

making (a) greater.

Example Question #4 : 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.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

Using the Binomial Theorem, if  is expanded, the  term is 

.

This makes  the coefficient of .

We compare the values of this expression at  for both  and .

 

(a)  If  and , the coefficient is 

.

This is the coefficient of .

(b) If  and , the coefficient is 

.

This is the coefficient of .

(b) is the greater quantity.

Example Question #5 : How To Find The Exponent Of Variables

Consider the expression 

Which is the greater quantity?

(a) The expression evaluated at 

(b) The expression evaluated at 

Possible Answers:

(b) is greater

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

Use the properties of powers to simplify the expression:

(a) If , then 

(b) If , then 

(b) is greater.

Example Question #8 : How To Find The Exponent Of Variables

Which of the following expressions is equivalent to 

 ?

Possible Answers:

None of the other answers is correct.

Correct answer:

None of the other answers is correct.

Explanation:

Use the square of a binomial pattern as follows:

This expression is not equivalent to any of the choices.

Example Question #6 : How To Find The Exponent Of Variables

Express   in terms of .

Possible Answers:

Correct answer:

Explanation:

 

, so

 

, so 

Example Question #10 : How To Find The Exponent Of Variables

. Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

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

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

By the Power of a Power Principle, 

Therefore, 

It follows that 

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