ISEE Upper Level Quantitative : How to multiply exponential variables

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

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Example Questions

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Example Question #931 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 and  are positive integers. Which is the greater quantity?

(A) 

(B) 

Possible Answers:

(B) is greater

(A) is greater

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

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

Since  and  are positive, 

 for all positive  and , making (B) greater.

Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 and  are negative integers. Which is the greater quantity?

(A) 

(B) 

Possible Answers:

(B) is greater

(A) is greater

(A) and (B) are equal

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

Correct answer:

(B) is greater

Explanation:

Since  and  are both negative, 

.

 for all negative  and , making (B) greater.

Example Question #933 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 and  are positive integers. Which is the greater quantity?

(A) 

(B) 

Possible Answers:

(A) and (B) are equal

(A) is greater

(B) is greater

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

Correct answer:

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

Explanation:

It is impossible to tell which is greater. 

Case 1: 

Then

and 

.

This makes (A) and (B) equal.

Case 2: 

Then

and 

.

This makes (A) the greater quantity.

Example Question #11 : How To Multiply Exponential Variables

 and  are positive integers greater than 1.

Which is the greater quantity?

(A) 

(B) 

Possible Answers:

(A) is greater

(A) and (B) are equal

(B) is greater

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

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute . Since  must be positive, and this problem is to compare  and  

and 

Since 2, , and  are positive, by closure, , and by the addition property of inequality,

Substituting back:

(A) is the greater quantity.

Example Question #934 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 and  are positive integers greater than 1.

Which is the greater quantity?

(A) 

(B)  

Possible Answers:

(A) is greater

(A) and (B) are equal

(B) is greater

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

Correct answer:

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

Explanation:

Case 1: 

Then

and  

This makes the quantities equal.

 

Case 2:

Then

and  

This makes (B) greater.

 

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

Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 and  are positive integers greater than 1.

Which is the greater quantity?

(A) 

(B) 

Possible Answers:

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

(A) is greater

(B) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute . The expressions to be compared are 

and 

Since  is positive, so is , and

Substituting back,

,

making (A) greater.

Example Question #937 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Factor:

Possible Answers:

The expression is a prime polynomial.

Correct answer:

Explanation:

We can rewrite as follows:

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

,

which, by distribution, becomes

Example Question #938 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

 is a positive number;  is the additive inverse of .

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

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

.

, as the difference of the squares of two expressions, can be factored as follows:

Since , it follows that 

Another consequence of  being the additive inverse of  is that

, so

 is positive, so  is as well.

It follows that .

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