# ISEE Upper Level Quantitative : Variables

## Example Questions

### Example Question #61 : Variables

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

(b) is greater.

(a) is greater.

The two quantities are equal.

It is impossible to tell from the information given.

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 #1 : Variables And Exponents

Which is greater?

(a)

(b)

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(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)

(b) is greater.

It is impossble to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(a) is greater.

Explanation:

Simplify the expression in (a):

Since

,

making (a) greater.

### Example Question #11 : Variables And Exponents

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

(b) is greater.

The two quantities are equal.

(a) is greater.

It is impossible to tell from the information given.

(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 #12 : Variables And Exponents

Consider the expression

Which is the greater quantity?

(a) The expression evaluated at

(b) The expression evaluated at

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(b) is greater

Explanation:

Use the properties of powers to simplify the expression:

(a) If , then

(b) If , then

(b) is greater.

### Example Question #13 : Variables And Exponents

Which of the following expressions is equivalent to

?

None of the other answers is correct.

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 #14 : Variables And Exponents

Express   in terms of .

Explanation:

, so

, so

### Example Question #15 : Variables And Exponents

. Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity

(a) is the greater quantity

Explanation:

By the Power of a Power Principle,

Therefore,

It follows that

### Example Question #11 : How To Find The Exponent Of Variables

is a real number such that . Which is the greater quantity?

(a)

(b) 11

(a) is the greater quantity

(b) is the greater quantity

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

(a) and (b) are equal

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

Explanation:

By the Power of a Power Principle,

Therefore,  is a square root of 121, of which there are two - 11 and . Since it is possible for a third (odd-numbered) power of a real number to be positive or negative, we cannot eliminate either possibility, so either

or

.

Therefore, we cannot determine whether  is less than 11 or equal to 11.