# SAT II Math I : Mathematical Relationships

## Example Questions

### Example Question #1 : Mathematical Relationships

Evaluate the expression.

Explanation:

Follow the correct order of operations: parenthenses, exponents, multiplication, division, addition, subtraction.

First, evaluate any terms in parenthesis.

Next, evaluate the exponent.

Divide.

### Example Question #2 : Mathematical Relationships

Evalute the expression:

Explanation:

Follow the correct order of operations: parentheses, exponents, multiplication, division, addition, subtraction. (This is typically abbreviated as PEMDAS. Note that both multiplication and division, and addition and subtraction, are equal to each other in terms of rank, so when both are present, solving the equation proceeds from left to right).

First, simplify anything in parentheses.

Next, simplify any terms with exponents.

Now, perform multiplication.

Since all we are left with is addition and subtraction, we perform simplification from left to right.

### Example Question #3 : Mathematical Relationships

Explanation:

In modulo 7 arithmetic, a number is congruent to the remainder of its division by 7.

Therefore, since  and ,

,

and the correct response is 3.

### Example Question #4 : Mathematical Relationships

Explanation:

To solve , make sure the digits are aligned with the correct placeholder.  It is also possible to add term by term.

### Example Question #5 : Mathematical Relationships

Evaluate: .

Explanation:

Step 1: Recall PEMDAS...

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Step 2: Perform the evaluation in separate pieces...

Step 3: Replace the values and keep the signs..

Step 4: Evaluate:

### Example Question #6 : Mathematical Relationships

Find the sum of the numbers:

Explanation:

Add the tens digits with the two as the carryover.

Combine this value with the ones digit of the first number.

### Example Question #7 : Mathematical Relationships

Evaluate:

Explanation:

Add the tens digits with the tens digit of the previous number as carryover.

Repeat the process with the hundreds digits.

Combine this number with the ones digits of the previous calculations.

### Example Question #4 : Mathematical Relationships

How many elements are in a set that has exactly 128 subsets?

None of the other responses is correct.

Explanation:

A set with  elements has  subsets.

Solve:

### Example Question #2 : Exponents And Logarithms

Rewrite as a single logarithmic expression:

Explanation:

Using the properties of logarithms

and ,

we simplify as follows:

Solve: