### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #61 : Variables

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

**Possible Answers:**

(b) is greater.

(a) is greater.

The two quantities are equal.

It is impossible to tell from the information given.

**Correct answer:**

The two quantities are equal.

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)

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

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:**

(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.

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

**Possible Answers:**

(b) is greater.

The two quantities are equal.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

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

**Possible Answers:**

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

**Correct answer:**

(b) is greater

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

?

**Possible Answers:**

None of the other answers is correct.

**Correct answer:**

None of the other answers is correct.

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 .

**Possible Answers:**

**Correct answer:**

, so

, so

### Example Question #15 : Variables And Exponents

. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(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

**Correct answer:**

(a) is the greater quantity

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

**Possible Answers:**

(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

**Correct answer:**

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

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.

### Example Question #12 : How To Find The Exponent Of Variables

**Possible Answers:**

**Correct answer:**

By the Power of a Product Principle,

Also, by the Power of a Power Principle

Therefore,

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