### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : How To Multiply Exponential Variables

Simplify:

**Possible Answers:**

**Correct answer:**

### Example Question #2 : How To Multiply Exponential Variables

Expand:

**Possible Answers:**

**Correct answer:**

A binomial can be cubed using the pattern:

Set

### Example Question #82 : Variables

Factor completely:

**Possible Answers:**

**Correct answer:**

A trinomial whose leading term has a coefficent other than 1 can be factored using the -method. We split the middle term using two numbers whose product is and whose sum is . These numbers are , so:

### Example Question #4 : How To Multiply Exponential Variables

Multiply:

**Possible Answers:**

**Correct answer:**

This can be achieved by using the pattern of difference of squares:

Applying the binomial square pattern:

### Example Question #5 : How To Multiply Exponential Variables

Simplify:

**Possible Answers:**

**Correct answer:**

The cube of a sum pattern can be applied here:

### Example Question #6 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

**Possible Answers:**

**Correct answer:**

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

### Example Question #2 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

**Possible Answers:**

**Correct answer:**

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

### Example Question #8 : How To Multiply Exponential Variables

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) 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)

(b)

(a) is the greater quantity.

### Example Question #9 : How To Multiply Exponential Variables

Which is the greater quantity?

(a)

(b) 8

**Possible Answers:**

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

**Correct answer:**

(a) is greater

Since ,

, so

making (a) greater.

### Example Question #10 : How To Multiply Exponential Variables

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

**Correct answer:**

It is impossible to tell from the information given.

We show that either polynomial can be greater by giving two cases:

Case 1:

Case 2:

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