### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Exponents

Simplify:

**Possible Answers:**

The correct answer is not among the other choices.

**Correct answer:**

Apply the power of a power property:

### Example Question #1 : Exponential Operations

Simplify the expression:

**Possible Answers:**

**Correct answer:**

Apply the power of a product rule, then apply the power of a power rule:

### Example Question #1 : How To Multiply Exponents

Which of the following expressions is equal to ?

**Possible Answers:**

The expression is undefined.

**Correct answer:**

Any nonzero number raised to the power of 0 is equal to 1.

### Example Question #1 : Exponents

Which quantity is greater?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

(a)

(b)

(b) is the greater quantity.

### Example Question #3 : Exponential Operations

is positive.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

**Correct answer:**

(a) is greater

Use the power of a power property:

(a)

(b)

Since , . Subsequently,

,

making (a) greater

### Example Question #1 : Exponents

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

**Correct answer:**

(a) and (b) are equal

The two quantities are equal.

### Example Question #3 : How To Multiply Exponents

Two quantities are given - one in Column A and the other in Column B. Compare the quantities in the two columns.

Assume, in both columns, that .

Column A Column B

**Possible Answers:**

The quantity in Column B is greater.

There is no way to determine the relationship between the columns.

The quantity in Column A is greater.

The quantities in both columns are equal.

**Correct answer:**

There is no way to determine the relationship between the columns.

Column A gives simplifies to give us , and Column B simplifies to give us . At first glance, Column B is greater, as it would be for all answers greater than 1. However, if , the two columns are equal. Furthermore, if is negative, or a fraction, Column A is greater. Thus, since we could arrive at all three answers by using different numbers, we cannot determine the answer conclusively.

### Example Question #1 : Exponents

Which of the following expressions is equivalent to

?

**Possible Answers:**

**Correct answer:**

Use the difference of squares pattern as follows:

### Example Question #2 : How To Multiply Exponents

Column A Column B

**Possible Answers:**

The quantity in Column B is greater.

No relationship between the columns can be determined.

The quantities in both columns are equal.

The quantity in Column A is greater.

**Correct answer:**

The quantity in Column A is greater.

Anything raised to zero is equal to 1. Therefore, Column A has to be greater because 1 is greater than 0.

### Example Question #10 : Exponential Operations

44,000,000 can be written in scientific notation as for some .

Which is the greater quantity?

(A)

(B) 8

**Possible Answers:**

(B) is greater

(A) and (B) are equal

(A) is greater

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

**Correct answer:**

(B) is greater

To write 44,000,000 in scientifc notation, write the implied decimal point after the final "0", then move it left until it is after the first nonzero digit (the first "4").

This requires a displacement of seven places, so

, and (B) is greater.