### All PSAT Math Resources

## Example Questions

### Example Question #71 : Exponents

If , what is the value of ?

**Possible Answers:**

**Correct answer:**

Using exponents, 27 is equal to 3^{3}. So, the equation can be rewritten:

3^{4x + 6} = (3^{3})^{2x}

3^{4x + 6} = 3^{6x}

When both side of an equation have the same base, the exponents must be equal. Thus:

4*x* + 6 = 6*x*

6 = 2*x*

*x* = 3

### Example Question #11 : Exponents

What is the value of such that ?

**Possible Answers:**

**Correct answer:**

We can solve by converting all terms to a base of two. 4, 16, and 32 can all be expressed in terms of 2 to a standard exponent value.

We can rewrite the original equation in these terms.

Simplify exponents.

Finally, combine terms.

From this equation, we can see that .

### Example Question #12 : How To Add Exponents

How many of the following base ten numbers have a base five representation of exactly four digits?

(A)

(B)

(C)

(D)

**Possible Answers:**

Two

One

None

Three

Four

**Correct answer:**

Three

A number in base five has powers of five as its place values; starting at the right, they are

The lowest base five number with four digits would be

in base ten.

The lowest base five number with five digits would be

in base ten.

Therefore, a number that is expressed as a four-digit number in base five must fall in the range

Three of the four numbers - all except 100 - fall in this range.

### Example Question #491 : Algebra

Solve for :

**Possible Answers:**

**Correct answer:**

Combining the powers, we get .

From here we can use logarithms, or simply guess and check to get .

### Example Question #501 : Algebra

If and are positive integers, and , then what is in terms of ?

**Possible Answers:**

**Correct answer:**

is equal to which is equal to . If we compare this to the original equation we get

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

If *m* and *n* are integers such that *m* < *n* < 0 and *m*^{2} – *n*^{2} = 7, which of the following can be the value of *m* + *n*?

I. –5

II. –7

III. –9

**Possible Answers:**

I, II and III only

II and III only

I and II only

I only

II only

**Correct answer:**

II only

*m* and *n* are both less than zero and thus negative integers, giving us *m*^{2} and *n*^{2} as perfect squares. The only perfect squares with a difference of 7 is 16 – 9, therefore m = –4 and n = –3.

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

**Possible Answers:**

**Correct answer:**

To simplify, we can rewrite the numerator using a common exponential base.

Now, we can factor out the numerator.

The eights cancel to give us our final answer.

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

If , then what is ?

**Possible Answers:**

**Correct answer:**

Follow the order of operations by solving the expression within the parentheses first.

Return to solve the original expression.

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

Solve:

**Possible Answers:**

**Correct answer:**

Subtract the denominator exponent from the numerator's exponent, since they have the same base.

### Example Question #12 : Exponents

5^{4 }/ 25 =

**Possible Answers:**

10

5^{4} / 5

50

5

25

**Correct answer:**

25

25 = 5 * 5 = 5^{2}. Then 5^{4} / 25 = 5^{4} / 5^{2}.

Now we can subtract the exponents because the operation is division. 5^{4} / 5^{2 }= 5^{4 – 2} = 5^{2} = 25. The answer is therefore 25.

Certified Tutor