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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

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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Example Question #128 : Multiplying And Dividing Exponents

Evaluate: $\frac{3^{15}}{27^{18}}$

Possible Answers:

$\frac{1}{9^3}$

$\frac{1}{3^{39}}$

$\frac{1}{9^{39}}$

$\frac{1}{9^{117}}$

$\frac{1}{3^9}$

Correct answer:

$\frac{1}{3^{39}}$

Explanation:

In order to simplify this, we will need to rewrite the base 27 as a common base of three in order to be able to subtract the exponents.

27 = 3^3

Rewrite the fraction.

$\frac{3^{15}}{27^{18}} = \frac{3^{15}}{3^{3(18)}} = \frac{3^{15}}{3^{54}}$

Subtract the exponents.

$3^{15-54} = 3^{-39}=\frac{1}{3^{39}}$

The answer is: $\frac{1}{3^{39}}$

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Example Question #131 : Simplifying Exponents

Divide: $\frac{3^{260}}{9^{132}}$

Possible Answers:

$\frac{1}{27}$

$\frac{1}{36}$

$\frac{1}{729}$

$\frac{1}{81}$

Correct answer:

$\frac{1}{81}$

Explanation:

In order to subtract the exponents, we will need similar bases.

Rewrite the nine as a base of three. Nine is equivalent to 3^2 .

The expression becomes:

$\frac{3^{260}}{9^{132}}= \frac{3^{260}}{3^{2(132)}}=\frac{3^{260}}{3^{264}}$

Now that the bases are equal, we can subtract the exponents and simplify.

$3^{260-264} = 3^{-4}=\frac{1}{3^4} = \frac{1}{81}$

The answer is: $\frac{1}{81}$

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Example Question #132 : Multiplying And Dividing Exponents

Simplify: $2^{18}\cdot 8^{500}$

Possible Answers:

$16^{9000}$

$\textup{The answer cannot be determined.}$

$2^{1518}$

$16^{518}$

$8^{4500}$

Correct answer:

$2^{1518}$

Explanation:

In order to simplify this, we will need to convert the base eight to base two.

8=2^3

Rewrite the second term.

$2^{18}\cdot 8^{500} = 2^{18}\cdot (2^3)^{500} = 2^{18}\cdot2^{1500}$

Because the bases are now similar, we can add the exponents.

$2^{18}\cdot2^{1500} = 2^{1500+18} = 2^{1518}$

The answer is: $2^{1518}$

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Example Question #133 : Multiplying And Dividing Exponents

Divide the exponents: $\frac{8^{100}}{16^{350}}$

Possible Answers:

$\frac{1}{2^{600}}$

$\frac{1}{2^{250}}$

$\frac{1}{2^{1100}}$

$\frac{1}{16^{250}}$

$\frac{1}{4^{450}}$

Correct answer:

$\frac{1}{2^{1100}}$

Explanation:

Rewrite the numerator and denominator using the base power of two.

$\frac{8^{100}}{16^{350}} = \frac{2^{3(100)}}{2^{4(350)}}$

Use the product rule of exponents to simplify the powers.

$\frac{2^{300}}{2^{1400}}$

Now that the bases are similar, the exponents on the numerator and denominator can be subtracted.

$\frac{2^{300}}{2^{1400}} = 2^{300-1400} = 2^{-1100} = \frac{1}{2^{1100}}$

The answer is: $\frac{1}{2^{1100}}$

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Example Question #134 : Multiplying And Dividing Exponents

Multiply the terms: $(2^{42})^2\cdot 2^{24}$

Possible Answers:

$2^{68}$

$4^{68}$

$2^{108}$

$2^{106}$

$4^{108}$

Correct answer:

$2^{108}$

Explanation:

Use the power rule of exponents to simplify. Multiply the power of 42 with the power of two outside the parentheses.

$(2^{42})^2\cdot 2^{24} = 2^{84}\cdot 2^{24}$

When the powers of the same bases are multiplied, the powers can be added.

$2^{84+24} = 2^{108}$

The answer is: $2^{108}$

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Example Question #135 : Multiplying And Dividing Exponents

Divide: $\frac{8^{200}}{4^{100}}$

Possible Answers:

$2^{50}$

$2^{200}$

$2^{250}$

$2^{400}$

$4^{200}$

Correct answer:

$2^{400}$

Explanation:

In order to be able to subtract the exponents, we will need to have similar bases in the numerator and denominator.

8=2^3

4=2^2

Replace the values and simplify.

$\frac{8^{200}}{4^{100}} = \frac{2^{3(200)}}{2^{2(100)}} = \frac{2^{600}}{2^{200}}$

Subtract the exponents.

The answer is: $2^{400}$

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Example Question #136 : Multiplying And Dividing Exponents

Divide: $\frac{25^{30}}{5^{90}}$

Possible Answers:

$5^{30}$

$25^{3}$

$\frac{1}{5^{30}}$

$\frac{1}{5^{60}}$

$\frac{1}{25^{3}}$

Correct answer:

$\frac{1}{5^{30}}$

Explanation:

Change the base of the numerator so that it matches the denominator.

25= 5^2

We can rewrite the numerator as five squared.

$\frac{25^{30}}{5^{90}} = \frac{(5^2)^{30}}{5^{90}}$

Use the product rule of exponents to simplify the numerator.

$\frac{(5^2)^{30}}{5^{90}} = \frac{5^{60}}{5^{90}}$

Subtract the powers.

$5^{60-90} = 5^{-30}$

Rewrite the negative exponent as a fraction.

The answer is: $\frac{1}{5^{30}}$

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Example Question #3591 : Algebra Ii

Simplify: $125(5^{50})(25^{100})$

Possible Answers:

$5^{30000}$

$125^{50}$

$25^{153}$

$\textup{Cannot be determined.}$

$5^{253}$

Correct answer:

$5^{253}$

Explanation:

Rewrite each term as a base of five.

125 = 5^3

5=5^1

25=5^2

Replace the terms.

$125(5^{50})(25^{100}) = 5^3(5^{1(50)})(5^{2(100)})= (5^3)(5^{50})(5^{200})$

Now that the similar bases with exponents are multiplied, the powers can be added.

$(5^3)(5^{50})(5^{200}) = (5^{3+50+200}) = 5^{253}$

The answer is: $5^{253}$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #128 : Multiplying And Dividing Exponents

Example Question #454 : Exponents

Example Question #128 : Multiplying And Dividing Exponents

Example Question #131 : Simplifying Exponents

Example Question #132 : Multiplying And Dividing Exponents

Example Question #133 : Multiplying And Dividing Exponents

Example Question #134 : Multiplying And Dividing Exponents

Example Question #135 : Multiplying And Dividing Exponents

Example Question #136 : Multiplying And Dividing Exponents

Example Question #3591 : Algebra Ii

All Algebra II Resources

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