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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #3601 : Algebra Ii

Simplify: x^5*x^6 $\displaystyle x^5*x^6$

Possible Answers:

$x^{11}$ $\displaystyle x^{11}$

$2x^{30}$ $\displaystyle 2x^{30}$

$x^{12}$ $\displaystyle x^{12}$

$2x^{11}$ $\displaystyle 2x^{11}$

$x^{30}$ $\displaystyle x^{30}$

Correct answer:

$x^{11}$ $\displaystyle x^{11}$

Explanation:

When multiplying exponents with the same base, we just keep the base the same and add the exponents.

$x^5*x^6=x^{5+6}=x^{11}$ $\displaystyle x^5*x^6=x^{5+6}=x^{11}$

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Example Question #132 : Simplifying Exponents

Simplify: $x^{12}*x^{-8}$ $\displaystyle x^{12}*x^{-8}$

Possible Answers:

$2x^{4}$ $\displaystyle 2x^{4}$

$x^{-4}$ $\displaystyle x^{-4}$

$x^{-2}$ $\displaystyle x^{-2}$

x^4 $\displaystyle x^4$

$x^{-96}$ $\displaystyle x^{-96}$

Correct answer:

x^4 $\displaystyle x^4$

Explanation:

When multiplying exponents with the same base, we just keep the base the same and add the exponents.

$x^{12}*x^{-8}=x^{12+(-8)}=x^4$ $\displaystyle x^{12}*x^{-8}=x^{12+(-8)}=x^4$

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Example Question #131 : Simplifying Exponents

Simplify: $(\frac{x}{y})^4*(\frac{y^2}{x^{-3}})^2$ $\displaystyle (\frac{x}{y})^4*(\frac{y^2}{x^{-3}})^2$

Possible Answers:

$\frac{x^7}{y}$ $\displaystyle \frac{x^7}{y}$

$x^{10}y^3$ $\displaystyle x^{10}y^3$

$x^{10}$ $\displaystyle x^{10}$

$\frac{x}{y}$ $\displaystyle \frac{x}{y}$

$\frac{x^7}{y^3}$ $\displaystyle \frac{x^7}{y^3}$

Correct answer:

$x^{10}$ $\displaystyle x^{10}$

Explanation:

We apply the exponents first before simplifying the fractions.

$(\frac{x}{y})^4*(\frac{y^2}{x^{-3}})^2=\frac{x^4}{y^4}*\frac{y^4}{x^{-6}}$ $\displaystyle (\frac{x}{y})^4*(\frac{y^2}{x^{-3}})^2=\frac{x^4}{y^4}*\frac{y^4}{x^{-6}}$ The y^4 $\displaystyle y^4$ cancels and we have $\frac{x^4}{x^{-6}}$ $\displaystyle \frac{x^4}{x^{-6}}$ . When dividing exponents, we subtract the exponents and keep the base the same.

$\frac{x^4}{x^{-6}}=x^{4-(-6)}=x^{10}$ $\displaystyle \frac{x^4}{x^{-6}}=x^{4-(-6)}=x^{10}$

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Example Question #3602 : Algebra Ii

Simplify: $\frac{x^7}{x^8}$ $\displaystyle \frac{x^7}{x^8}$

Possible Answers:

$x^{15}$ $\displaystyle x^{15}$

$\displaystyle x$

$-\frac{1}{x}$ $\displaystyle -\frac{1}{x}$

$\frac{1}{x^2}$ $\displaystyle \frac{1}{x^2}$

$\frac{1}{x}$ $\displaystyle \frac{1}{x}$

Correct answer:

$\frac{1}{x}$ $\displaystyle \frac{1}{x}$

Explanation:

When dividing exponents, we subtract the exponents and keep the base the same.

$\frac{x^7}{x^8}=x^{7-8}=x^{-1}$ $\displaystyle \frac{x^7}{x^8}=x^{7-8}=x^{-1}$ We know with negative exponents, it's expressed as one over the positive exponent.

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

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Example Question #472 : Exponents

Simplify: $x^5*(\frac{5}{x^4})^2*\frac{x^8}{10}$ $\displaystyle x^5*(\frac{5}{x^4})^2*\frac{x^8}{10}$

Possible Answers:

$\frac{5x^4}{2}$ $\displaystyle \frac{5x^4}{2}$

$\frac{x^9}{2}$ $\displaystyle \frac{x^9}{2}$

x^5 $\displaystyle x^5$

$\frac{5x^9}{2}$ $\displaystyle \frac{5x^9}{2}$

$\frac{5x^5}{2}$ $\displaystyle \frac{5x^5}{2}$

Correct answer:

$\frac{5x^5}{2}$ $\displaystyle \frac{5x^5}{2}$

Explanation:

Let's apply the exponents to the parentheses first and then simplify.

$x^5*(\frac{5}{x^4})^2*\frac{x^8}{10}=x^5*\frac{25}{x^8}*\frac{x^8}{10}$ $\displaystyle x^5*(\frac{5}{x^4})^2*\frac{x^8}{10}=x^5*\frac{25}{x^8}*\frac{x^8}{10}$ The x^8 $\displaystyle x^8$ cancels and the numbers can be reduced by $\displaystyle 5$ .

We finally get: $\frac{5x^5}{2}$ $\displaystyle \frac{5x^5}{2}$ .

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Example Question #3603 : Algebra Ii

Simplify and express as exponents: $\frac{x^4}{y^7}*x^{-8}$ $\displaystyle \frac{x^4}{y^7}*x^{-8}$

Possible Answers:

(xy)^4 $\displaystyle (xy)^4$

$x^{-4}y^{-7}$ $\displaystyle x^{-4}y^{-7}$

$x^{-4}y^7$ $\displaystyle x^{-4}y^7$

(xy)^7 $\displaystyle (xy)^7$

x^4y^7 $\displaystyle x^4y^7$

Correct answer:

$x^{-4}y^{-7}$ $\displaystyle x^{-4}y^{-7}$

Explanation:

Let's rewrite this as just exponents. Remember we can breakup $\frac{x^4}{y^7}=x^4*\frac{1}{y^7}$ $\displaystyle \frac{x^4}{y^7}=x^4*\frac{1}{y^7}$ . $\frac{x^4}{y^7}*x^{-8}=\frac{1}{y^7}*x^4*x^{-8}=y^{-7}*x^{-4}$ $\displaystyle \frac{x^4}{y^7}*x^{-8}=\frac{1}{y^7}*x^4*x^{-8}=y^{-7}*x^{-4}$

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Example Question #3604 : Algebra Ii

Simplify: $\frac{5^{2x}}{5^{2x+4}}$ $\displaystyle \frac{5^{2x}}{5^{2x+4}}$

Possible Answers:

$-\frac{1}{5^4}$ $\displaystyle -\frac{1}{5^4}$

-5^4 $\displaystyle -5^4$

5^4 $\displaystyle 5^4$

$\displaystyle 1$

$\frac{1}{5^4}$ $\displaystyle \frac{1}{5^4}$

Correct answer:

$\frac{1}{5^4}$ $\displaystyle \frac{1}{5^4}$

Explanation:

When dividing exponents, we subtract the exponents and keep the base the same.

$\frac{5^{2x}}{5^{2x+4}}=5^{2x-(2x+4)}=5^{2x-2x-4}=5^{-4}=\frac{1}{5^4}$ $\displaystyle \frac{5^{2x}}{5^{2x+4}}=5^{2x-(2x+4)}=5^{2x-2x-4}=5^{-4}=\frac{1}{5^4}$ We know with negative exponents, it's expressed as one over the positive exponent.

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Example Question #3605 : Algebra Ii

Simplify: $\frac{7^{8x+4}}{7^{7x-2}}$ $\displaystyle \frac{7^{8x+4}}{7^{7x-2}}$

Possible Answers:

$\frac{1}{7^{x+2}}$ $\displaystyle \frac{1}{7^{x+2}}$

$7^{x+6}$ $\displaystyle 7^{x+6}$

$\frac{1}{7^{x+6}}$ $\displaystyle \frac{1}{7^{x+6}}$

$7^{x+2}$ $\displaystyle 7^{x+2}$

$-\frac{1}{7^{x+6}}$ $\displaystyle -\frac{1}{7^{x+6}}$

Correct answer:

$7^{x+6}$ $\displaystyle 7^{x+6}$

Explanation:

When dividing exponents, we subtract the exponents and keep the base the same.

$\frac{7^{8x+4}}{7^{7x-2}}=7^{8x+4-(7x-2)}=7^{8x+4-7x+2}=7^{x+6}$ $\displaystyle \frac{7^{8x+4}}{7^{7x-2}}=7^{8x+4-(7x-2)}=7^{8x+4-7x+2}=7^{x+6}$

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Example Question #476 : Exponents

Simplify: $\frac{a+b}{a^2+2ab+b^2}$ $\displaystyle \frac{a+b}{a^2+2ab+b^2}$

Possible Answers:

a+b $\displaystyle a+b$

a-b $\displaystyle a-b$

$\frac{1}{2ab}$ $\displaystyle \frac{1}{2ab}$

$\frac{1}{a+ab+b}$ $\displaystyle \frac{1}{a+ab+b}$

$\frac{1}{a+b}$ $\displaystyle \frac{1}{a+b}$

Correct answer:

$\frac{1}{a+b}$ $\displaystyle \frac{1}{a+b}$

Explanation:

Although it seems like we can't simplify anything, we do know that $\displaystyle a^2+2ab+b^2=(a+b)^2$ . Therefore we have:

$\frac{a+b}{a^2+2ab+b^2}=\frac{a+b}{(a+b)^2}$ $\displaystyle \frac{a+b}{a^2+2ab+b^2}=\frac{a+b}{(a+b)^2}$ . Now we can divide the exponents to get $(a+b)^{-1}=\frac{1}{a+b}$ $\displaystyle (a+b)^{-1}=\frac{1}{a+b}$

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Example Question #477 : Exponents

Simplify: $(\frac{x}{y})^{7n}*(\frac{x}{y^{-1}})^{2n}*\frac{y^{5n}}{x^{5n+2}}$ $\displaystyle (\frac{x}{y})^{7n}*(\frac{x}{y^{-1}})^{2n}*\frac{y^{5n}}{x^{5n+2}}$

Possible Answers:

$x^{4n+2}$ $\displaystyle x^{4n+2}$

$\frac{x^{4n}}{y^{2n}}$ $\displaystyle \frac{x^{4n}}{y^{2n}}$

$\frac{x^n}{y^n}$ $\displaystyle \frac{x^n}{y^n}$

$\frac{x^{4n-2}}{y^n}$ $\displaystyle \frac{x^{4n-2}}{y^n}$

$x^{4n-2}$ $\displaystyle x^{4n-2}$

Correct answer:

$x^{4n-2}$ $\displaystyle x^{4n-2}$

Explanation:

Let's apply the exponential operation before we simplify.

$(\frac{x}{y})^{7n}*(\frac{x}{y^{-1}})^{2n}*\frac{y^{5n}}{x^{5n+2}}=\frac{x^{7n}}{y^{7n}}*x^{2n}y^{2n}*\frac{y^{5n}}{x^{5n+2}}$ $\displaystyle (\frac{x}{y})^{7n}*(\frac{x}{y^{-1}})^{2n}*\frac{y^{5n}}{x^{5n+2}}=\frac{x^{7n}}{y^{7n}}*x^{2n}y^{2n}*\frac{y^{5n}}{x^{5n+2}}$ In the numerator, the $y^{2n}, y^{5n}$ $\displaystyle y^{2n}, y^{5n}$ become $y^{7n}$ $\displaystyle y^{7n}$ and cancels with the denominator in the left fraction.

We now have: $x^{7n}*x^{2n}*\frac{1}{x^{5n+2}}$ $\displaystyle x^{7n}*x^{2n}*\frac{1}{x^{5n+2}}$ . By combining the top and applying the division rule of exponents, we get:

$x^{7n}*x^{2n}*\frac{1}{x^{5n+2}}=\frac{x^{9n}}{x^{5n+2}}=x^{9n-(5n+2)}=x^{9n-5n-2}=x^{4n-2}$ $\displaystyle x^{7n}*x^{2n}*\frac{1}{x^{5n+2}}=\frac{x^{9n}}{x^{5n+2}}=x^{9n-(5n+2)}=x^{9n-5n-2}=x^{4n-2}$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3601 : Algebra Ii

Example Question #132 : Simplifying Exponents

Example Question #131 : Simplifying Exponents

Example Question #3602 : Algebra Ii

Example Question #472 : Exponents

Example Question #3603 : Algebra Ii

Example Question #3604 : Algebra Ii

Example Question #3605 : Algebra Ii

Example Question #476 : Exponents

Example Question #477 : Exponents

All Algebra II Resources

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