### All ISEE Upper Level Math Resources

## Example Questions

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

Simplify the following:

**Possible Answers:**

None of the other answers

**Correct answer:**

To multiply variables with exponents, add the exponents. With multiple variables, simply add the exponents for each different variable.

Simplified:

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

Simplify the following:

**Possible Answers:**

None of the other answers

**Correct answer:**

To multiply variables with exponents, add the exponents. When there are constants mixed in, multiply the constants separately and put back in the final result:

### Example Question #3 : How To Multiply Exponential Variables

Simplify the following:

**Possible Answers:**

**Correct answer:**

To multiply variables with exponents, add the exponents. So,

A longer way would be to write out all the multiplies the exponent tells us to do. This is a little clearer on why adding the exponents works but takes longer and isn't necessary once you understand the process.

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

Factor completely:

**Possible Answers:**

**Correct answer:**

is the greatest common factor of each term, so distribute it out:

We try to factor by finding two integers with product 4 and sum . However, both of our possible factor pairs fail, since and .

is the complete factorization.

### Example Question #5 : 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 #6 : How To Multiply Exponential Variables

Factor completely:

**Possible Answers:**

**Correct answer:**

The greatest common factor of the terms in is , so factor that out:

Since all factors here are linear, this is the complete factorization.

### Example Question #7 : How To Multiply Exponential Variables

Exponentiate:

**Possible Answers:**

**Correct answer:**

The cube of a sum pattern can be applied here:

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

Write in expanded form:

**Possible Answers:**

**Correct answer:**

The cube of a sum pattern can be applied here:

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

Factor completely:

**Possible Answers:**

The expression cannot be factored.

**Correct answer:**

The expression cannot be factored.

A trinomial with leading coefficient can be factored by looking for two integers to fill in the boxes:

.

The numbers should have product 20 and sum . However, all of the possible factor pairs fail:

The polynomial is prime.

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

Factor completely:

**Possible Answers:**

**Correct answer:**

can be seen to be a perfect square trinomial by taking the square root of the first and last terms, multiplying their product by 2, then comparing it to the second term:

Therefore,