# ISEE Upper Level Math : Numbers and Operations

## Example Questions

### Example Question #8 : Exponents

Define  as follows:

Evaluate .

Explanation:

### Example Question #9 : Exponents

Simplify the expresseion:

Explanation:

Variable terms can be combined by adding and subtracting if and only of they are like - that is, if each exponent of each variable is the same. In the given expression, no two exponents are the same. The terms cannot be combined, and the expression is already simplified.

### Example Question #10 : Exponents

Define a function  as follows:

Evaluate .

Explanation:

### Example Question #61 : Numbers And Operations

Define a function .

Evaluate

Explanation:

### Example Question #62 : Numbers And Operations

Define  and .

Evaluate .

Undefined

Explanation:

, by definition, so

Evaluate  and  separately:

### Example Question #63 : Numbers And Operations

Define  as follows:

Evaluate .

Explanation:

### Example Question #64 : Numbers And Operations

Explanation:

Subtract the numbers inside the parentheses first.  This leaves you with 3, which you then raise to the 3rd power:

### Example Question #65 : Numbers And Operations

Evaluate:

The expression is undefined.

Explanation:

Any nonzero number taken to the power of 0 is equal to 1, so

### Example Question #66 : Numbers And Operations

Multiply:

Explanation:

Use the FOIL method:

### Example Question #67 : Numbers And Operations

Evaluate:

Explanation:

Based on the power rule for exponents we can write:

That means; to raise a power to a power we need to multiply the exponents. In addition, based on the product rule for exponents in order to multiply two exponential terms with the same base, we need to add their exponents:

So we can write: