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## Example Questions

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

(b * b^{4 }* b^{7})^{1/2}/(b^{3} * b^{x}) = b^{5}

If b is not negative then x = ?

**Possible Answers:**

–2

1

7

–1

**Correct answer:**

–2

Simplifying the equation gives b^{6}/(b^{3+x}) = b^{5}.

In order to satisfy this case, x must be equal to –2.

### Example Question #22 : Exponents

If〖7/8〗^{n}= √(〖7/8〗^{5}),then what is the value of n?

**Possible Answers:**

^{2}/_{5}

√5

25

^{1}/_{5}

^{5}/_{2}

**Correct answer:**

^{5}/_{2}

7/8 is being raised to the 5th power and to the ^{1}/_{2} power at the same time. We multiply these to find n.

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

Quantity A:

(0.5)^{3}(0.5)^{3}

Quantity B:

(0.5)^{7}

**Possible Answers:**

The relationship cannot be determined from the information given.

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

**Correct answer:**

Quantity A is greater.

When we have two identical numbers, each raised to an exponent, and multiplied together, we add the exponents together:

^{}x^{a}x^{b} = x^{a+b}

This means that (0.5)^{3}(0.5)^{3} = (0.5)^{3+3} = (0.5)^{6}

Because 0.5 is between 0 and 1, we know that when it is multipled by itself, it decreases in value. Example: 0.5 * 0.5 = 0.25. 0.5 * 0.5 * 0.5 = 0.125. Etc.

Thus, (0.5)^{6} > (0.5)^{7}

### Example Question #2 : How To Multiply Exponents

For the quantities below, x<y and x and y are both integers.

Please elect the answer that describes the relationship between the two quantities below:

Quantity A

x^{5}y^{3}

Quantity B

x^{4}y^{4}

**Possible Answers:**

The relationship cannot be determined from the information provided.

Quantity B is greater.

The quantities are equal.

Quantity A is greater.

**Correct answer:**

The relationship cannot be determined from the information provided.

Answer: The relationship cannot be determined from the information provided.

Explanation: The best thing to do here is to notice that quantity A is composed of two complex terms with odd exponents. Odd powers result in negative results when their base is negative. Thus quantity A will be negative when either x or y (but not both) is negative. Otherwise, quantity A will be positive. Quantity B, however, has two even exponents, meaning that it will always be positive. Thus, sometimes Quantity A will be greater and sometimes Quantity B will be greater. Thus the answer is that the relationship cannot be determined.

### Example Question #3 : How To Multiply Exponents

Simplify: (x^{3} * 2x^{4} * 5y + 4y^{2} + 3y^{2})/y

**Possible Answers:**

None of the other answers

10x^{7}y + 7y^{2}

10x^{7} + 7y^{3}

10x^{11} + 7y^{3}

10x^{7} + 7y^{}

**Correct answer:**

10x^{7} + 7y^{}

Let's do each of these separately:

x^{3} * 2x^{4} * 5y = 2 * 5 * x^{3 }* x^{4 }* y = 10 * x^{7} * y = 10x^{7}y

4y^{2} + 3y^{2} = 7y^{2}

Now, rewrite what we have so far:

(10x^{7}y + 7y^{2})/y

There are several options for reducing this. Remember that when we divide, we can "distribute" the denominator through to each member. That means we can rewrite this as:

(10x^{7}y)/y + (7y^{2})/y

Subtract the y exponents values in each term to get:

10x^{7} + 7y

### Example Question #4 : How To Multiply Exponents

Quantitative Comparison

Quantity A: *x*^{3}/3

Quantity B: (*x*/3)^{3}

**Possible Answers:**

The two quantities are equal.

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined from the information given.

**Correct answer:**

The relationship cannot be determined from the information given.

First let's look at Quantity B:

(*x*/3)^{3} = *x*^{3}/27. Now both columns have an *x*^{3 }so we can cancel it from both terms. Therefore we're now comparing 1/3 in Quantity A to 1/27 in Quantity B. 1/3 is the larger fraction so Quantity A is greater.

However, if , then the two quantities would both equal 0. Thus, since the two quantities can have different relationships based on the value of , we cannot determine the relationship from the information given.

### Example Question #5 : How To Multiply Exponents

**Quantitative Comparison: **Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.

**Quantity A Quantity B**

(2^{3 })^{2} (2^{2 })^{3 }

^{ }

**Possible Answers:**

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

**Correct answer:**

The two quantities are equal.

The two quantites are equal. To take the exponent of an exponent, the two exponents should be multiplied.

(2^{3 })^{2 }or 2^{3*2} = 64

(2^{2 })^{3 }or 2^{2*3} = 64

Both quantities equal 64, so the two quantities are equal.

### Example Question #25 : Exponents

Compare and .

**Possible Answers:**

The answer cannot be determined from the information given.

**Correct answer:**

To compare these expressions more easily, we'll change the first expression to have in front. We'll do this by factoring out 25 (that is, ) from 850, then using the fact that .

When we combine like terms, we can see that . The two terms are therefore both equal to the same value.

### Example Question #26 : Exponents

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

is always equal to ; therefore, 5 raised to 4 times 5 raised to 5 must equal 5 raised to 9.

is always equal to . Therefore, 5 raised to 9, raised to 20 must equal 5 raised to 180.

### Example Question #26 : Exponents

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

First, multiply inside the parentheses: .

Then raise to the 7th power: .

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