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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #116 : Multiplying And Dividing Exponents

Simplify:

$\frac{16x^4y^6z^{10}}{40x^8y^2z^5}$

Possible Answers:

$\frac{2y^4z^5}{16x^4}$

$\frac{y^4z^5}{5x^4}$

$\frac{2y^3z^5}{5x^4}$

$\frac{2y^4z^5}{5x^2}$

$\frac{2y^4z^5}{5x^4}$

Correct answer:

$\frac{2y^4z^5}{5x^4}$

Explanation:

Remember, when simplifying exponents while dividing, subtract exponents. Simplify the like terms separately:

$\frac{16}{40}=\frac{2}{5}$ ,

$\frac{x^4}{x^8}=\frac{1}{x^4}$ ,

$\frac{y^6}{y^2}=y^4$ ,

$\frac{z^{10}}{z^5}=z^5$ .

Then, put all of those together to get your final answer:

$\frac{2y^4z^5}{5x^4}$ .

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Example Question #451 : Exponents

$9x\cdot (2x)^3$

Possible Answers:

72x^4

-72x^4

68x^2

72x^3

54x^4

Correct answer:

72x^4

Explanation:

First, simplify the second expression. When there is an exponent outside the parentheses, it must be distributed to every term in the parentheses:

(2x)^3=8x^3 .

Then multiply the two terms:

$9x\cdot 8x^3=72x^4$ .

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Example Question #122 : Multiplying And Dividing Exponents

Divide the exponents: $\frac{2^7}{4^{10}}$

Possible Answers:

4^4

$\frac{1}{8}$

$(\frac{1}{2})^{13}$

$\frac{1}{2^{13}}$

Correct answer:

$\frac{1}{2^{13}}$

Explanation:

In order to divide the exponents, we must have the same bases.

Convert the base 4 to 2 by rewriting the four as two squared.

$\frac{2^7}{4^{10}}=\frac{2^7}{2^{2(10)}}= \frac{2^7}{2^{20}}$

Now that the bases are the same, subtract the power on the numerator with the denominator.

$2^{7-20}= 2^{-13} = \frac{1}{2^{13}}$

The answer is: $\frac{1}{2^{13}}$

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Example Question #452 : Exponents

Simplify: $9^9\div9^{-9}$

Possible Answers:

$9^{-81}$

$9^{-18}$

$9^{18}$

Correct answer:

$9^{18}$

Explanation:

When dividing exponents with the same base, we subtract the exponents while keeping the base the same.

$9^9\div9^{-9}=9^{9-(-9)}=9^{18}$

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Example Question #124 : Multiplying And Dividing Exponents

Simplify:

$\frac{64x^{-3}y^5z^9}{80x^2y^{-2}z^3}$

Possible Answers:

$\frac{4y^7z^6}{5x^5}$

$\frac{4y^7z^3}{5x^5}$

$\frac{y^7z^6}{5x^5}$

$\frac{4y^7z^6}{x^5}$

$\frac{2y^7z^6}{5x^5}$

Correct answer:

$\frac{4y^7z^6}{5x^5}$

Explanation:

To simplify a fraction like this, I look at the like terms separately and simplify those, putting everything together at the end.

First,

$\frac{64}{80}=\frac{4}{5}$ .

Then,

$\frac{x^{-3}}{x^2}=\frac{1}{x^5}$ .

$\frac{y^{5}}{y^{-2}}=y^7$ .

$\frac{z^9}{z^3}=z^6$ .

Put those all together to get your answer:

$\frac{4y^7z^6}{5x^5}.$

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Example Question #125 : Multiplying And Dividing Exponents

$4x\cdot (-2x^3)\cdot (x^{-2})$

Possible Answers:

-8x^3

-8x^2

-2x^2

-8x

8x^2

Correct answer:

-8x^2

Explanation:

Multiply the like terms. Remember that when multiplying with exponents and bases are the same, add exponents.

Therefore,

4(-2)=-8

and

$x(x^3)(x^{-2})=x^2$ .

Thus, your answer is:

-8x^2.

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Example Question #453 : Exponents

Simplify:

$\frac{x^7y^4+x^8y^3+x^3y^7+x^3}{x^3y^3}$

Possible Answers:

$x^4y^1+x^5+y^4+y^{3}$

None of these

$x^1y^4+y^5+x^4+y^{-3}$

$x^4y^1+x^5+y^4+y^{-3}$

Correct answer:

$x^4y^1+x^5+y^4+y^{-3}$

Explanation:

When dividing variables with exponents, the exponents are subtracted

Thus

$\frac{x^7y^4+x^8y^3+x^3y^7+x^3}{x^3y^3}$

becomes

$x^4y^1+x^5+y^4+y^{-3}$

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Example Question #127 : Multiplying And Dividing Exponents

Simplify: $49^8\div7^{-8}$

Possible Answers:

$7^{24}$

$\frac{1}{7}$

$7^{-64}$

$\frac{1}{2}$

Correct answer:

$7^{24}$

Explanation:

Although the bases are not the same, we know that 49=7^2 . Therefore, $49^8=(7^2)^8=7^{16}$ . With the same bases, we now can subtract the exponents while keeping the base the same.

$7^{16}\div7^{-8}=7^{16-(-8)}=7^{24}$

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Example Question #128 : Multiplying And Dividing Exponents

Multiply the following exponents: $4^{30}\cdot 16^{12}$

Possible Answers:

$4^{132}$

$64^{18}$

$4^{54}$

$\textup{The answer does not exist.}$

$20^{42}$

Correct answer:

$4^{54}$

Explanation:

Notice that we can rewrite the second term in terms of the base of four. Sixteen is equal to four squared.

16= 4^2

We can replace this term with 16 in order to multiply, and then add the exponents.

$4^{30}\cdot 16^{12} =4^{30}\cdot 4^{2(12)} =4^{30}\cdot 4^{24}$

According to the rule of exponents, whenever powers of a similar base are multiplied, the exponents can be added.

$x^A \cdot x^B = x^{( A + B )}$

The answer is: $4^{54}$

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Example Question #454 : Exponents

Divide: $\frac{4^{34}}{4^{36}}$

Possible Answers:

$4^{70}$

$\frac{1}{16}$

Correct answer:

$\frac{1}{16}$

Explanation:

When dividing powers of a similar base, the exponents can be subtracted.

$\frac{4^{34}}{4^{36}} = 4^{34-36}=4^{-2}$

Simplify the negative exponent.

$x^{-n} = \frac{1}{x^n}$

$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

The answer is: $\frac{1}{16}$

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #116 : Multiplying And Dividing Exponents

Example Question #451 : Exponents

Example Question #122 : Multiplying And Dividing Exponents

Example Question #452 : Exponents

Example Question #124 : Multiplying And Dividing Exponents

Example Question #125 : Multiplying And Dividing Exponents

Example Question #453 : Exponents

Example Question #127 : Multiplying And Dividing Exponents

Example Question #128 : Multiplying And Dividing Exponents

Example Question #454 : Exponents

All Algebra II Resources

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