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Algebra 1 : Algebra 1

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #522 : Variables

Divide: $\frac{9ab^6}{36a^3b^6}$

Possible Answers:

$\frac{1}{27a^2}$

4a^2

$\textup{Cannot be solved.}$

$\frac{1}{4a^2}$

Correct answer:

$\frac{1}{4a^2}$

Explanation:

Rewrite the numerator and denominator in factors.

$\frac{9ab^6}{36a^3b^6} = \frac{9\times a \times b^6}{9 \times 4 \times a\times a^2 \times b^6}$

Cancel out the common terms in the numerator and denominator. Since all the terms in the numerator cancels out, the numerator becomes a one.

The answer is: $\frac{1}{4a^2}$

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Example Question #107 : Monomials

Simplify the following expression:

Which of the following expressions is equivalent to this one:

$\frac{4a^{3}b^{2}}{6a}\cdot\frac{8c^{4}b^{3}}{2b}$

Possible Answers:

$\frac{16a^2b^3c^4}{3}$

$\frac{8a^{2}b^{4}c^{4}}{3}$

$\frac{8a^{2}b^{3}c^{4}}{3}$

$\frac{32a^3b^3 c^4}{12}$

$\frac{8a^2bc^4}{6}$

Correct answer:

$\frac{8a^{2}b^{4}c^{4}}{3}$

Explanation:

First, simplify each term individually. Let's start with the term on the left. We can factor out from both the numerator and denominator.

$\frac{4a^{3}b^{2}}{6a}\rightarrow \frac{2a^2b^2}{3}$

Next, factor out from the numerator and denominator of the term on the right.

$\frac{8c^{4}b^{3}}{2b}\rightarrow \frac{4c^4b^2}{1}$

Combine and multiply. Remember that when we are multiplying terms with exponents, we need to add their exponential values. Likewise, when we divide exponents, we will subtract their exponential values.

$\frac{4a^3b^2}{6a}\cdot\frac{8c^4b^3}{2b} = \frac{2a^2b^2}{3}\cdot\frac{4c^4b^2}{1}$

$\frac{2a^2b^2}{3}\cdot\frac{4c^4b^2}{1}=\frac{(2a^2b^2)(4c^4b^2)}{3}$

Simplify.

$\frac{(2a^2b^2)(4c^4b^2)}{3}=\frac{8a^2b^4c^4}{3}$

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Example Question #1 : How To Multiply Monomial Quotients

Simplify:

$\frac{a^{2}b}{c^{4}}*\frac{c^{6}}{a^{5}b^{6}}$

Possible Answers:

$\frac{1}{a^{7}b^{7}c^{10}}$

$3a^{7}b^{7}c^{10}$

$\frac{a^{3}b^{5}}{c^{2}}$

$a^{\frac{2}{5}}b^{\frac{1}{7}}c^{\frac{3}{2}}$

$\frac{c^{2}}{a^{3}b^{5}}$

Correct answer:

$\frac{c^{2}}{a^{3}b^{5}}$

Explanation:

Since there are no like terms in the numerator or in the denominator, you can only combine ther terms on the numerator and denominator so that they are in one quotient.

$\frac{a^{2}b}{c^{4}}*\frac{c^{6}}{a^{5}b^{6}}=\frac{a^{2}bc^{6}}{a^{5}b^{6}c^{4}}$

Use the rules of exponents $\frac{x^{a}}{x^{b}}=x^{a-b}$ and $x^{-a}=\frac{1}{x^{a}}$ to further simplify the expression by combining the terms $a^{2}$ and $a^{5}$ , and $b^{6}$ , and $c^{6}$ and $c^{4}$ .

$\frac{a^{2}bc^{6}}{a^{5}b^{6}c^{4}}=a^{-3}b^{-5}c^{2}=\frac{c^{2}}{a^{3}b^{5}}$

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Example Question #531 : Variables

Simplify:

$\frac{x^{2}y^{2}}{4z}*\frac{16x^{4}yz^{2}}{y^{3}}$

Possible Answers:

$4x^{6}yz$

$4x^{6}y^{0}z^{2}$

$\frac{4x^{6}}{y^{6}z^{3}}$

$4x^{6}z$

$4x^{6}y^{6}z^{3}$

Correct answer:

$4x^{6}z$

Explanation:

Divide both integers by the GCF (4).

$\frac{x^{2}y^{2}}{4z}*\frac{16x^{4}yz^{2}}{y^{3}}=\frac{x^{2}y^{2}}{z}*\frac{4x^{4}yz^{2}}{y^{3}}$

Organize the variables so like terms are together in the numerator and in the denominator. Then use rules of exponents to combine $x^{2}$ and $x^{4}$ and $y^{2}$ and in the numerator. Remember the following exponent rule: $x^{a}*x^{b}=x^{a+b}$ ,

$\frac{x^{2}y^{2}}{z}*\frac{4x^{4}yz^{2}}{y^{3}}=\frac{4x^{2}x^{4}y^{2}yz^{2}}{y^{3}z}=\frac{4x^{6}y^{3}z^{2}}{y^{3}z}$

Since $y^{3}$ is in the numerator and the denominator, you can cancel it out.

$\frac{4x^{6}y^{3}z^{2}}{y^{3}z}=\frac{4x^{6}z^{2}}{z}$

Use the exponent rule $\frac{x^{a}}{x^{b}}=x^{a-b}$ to further simplify the expression by combining the terms $z^{2}$ and .

$\frac{4x^{6}z^{2}}{z}=4x^{6}z$

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Example Question #532 : Variables

Simplify:

$\frac{3a^{2}}{b^{4}}*\frac{4b}{a}*\frac{a^{2}b^{4}}{2}$

Possible Answers:

$6a^{4}b^{\frac{5}4{}}$

$\frac{6}{a^{5}b^{9}}$

$6a^{5}b^{9}$

$\frac{6}{a^{3}b}$

$6a^{3}b$

Correct answer:

$6a^{3}b$

Explanation:

Divide both 4 and 2 by the GCF (2) and organize the variables so like terms are together in the numerator and in the denominator. Multiply the integers 4 and 3 together.

$\frac{3a^{2}}{b^{4}}*\frac{4b}{a}*\frac{a^{2}b^{4}}{2}=\frac{3a^{2}}{b^{4}}*\frac{2b}{a}*\frac{a^{2}b^{4}}{1}=\frac{2*3*a^{2}a^{2}bb^{4}}{ab^{4}}=\frac{6a^{2}a^{2}bb^{4}}{ab^{4}}$

Then use rules of exponents to combine $a^{2}$ and $a^{2}$ and and $b^{4}$ in the numerator. Remember the following exponent rule: $x^{a}*x^{b}=x^{a+b}$ ,

$\frac{6a^{2}a^{2}bb^{4}}{ab^{4}}=\frac{6a^{4}b^{5}}{ab^{4}}$

Use the exponent rule $\frac{x^{a}}{x^{b}}=x^{a-b}$ to further simplify the expression by combining the terms $a^{4}$ and and $b^{5}$ and .

$\frac{6a^{4}b^{5}}{ab^{4}}=6a^{3}b$

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Example Question #533 : Variables

Simplify:

$\frac{12x^{3}y^{5}z^{12}}{3}*\frac{6xyz}{30x^{4}y^{3}z^{2}}$

Possible Answers:

$\small \frac{4y^{3}z^{11}}{5}$

$\small \frac{5x^{8}y^{9}z^{15}}{4}$

$\small \frac{5}{4y^{3}z^{11}}$

$\small \frac{4}{5x^{8}y^{9}z^{15}}$

$\small \frac{4}{5y^{9}z^{15}}$

Correct answer:

$\small \frac{4y^{3}z^{11}}{5}$

Explanation:

In the first quotient, divide 12 and 3 by the GCF (3). In the second quotient, divide 6 and 30 by the GCF (6). You could also divide all the intergers by 2, but it would take longer to simplify since you would end up with larger numbers.

$\frac{12x^{3}y^{5}z^{12}}{3}*\frac{6xyz}{30x^{4}y^{3}z^{2}}=4x^{3}y^{5}z^{12}*\frac{xyz}{5x^{4}y^{3}z^{2}}$

Organize the variables so like terms are together in the numerator and in the denominator. Then use rules of exponents to combine $x^{3}$ and and $y^{5}$ and and $z^{12}$ and in the numerator. Remember the following exponent rule: $x^{a}*x^{b}=x^{a+b}$ ,

$\small 4x^{3}y^{5}z^{12}*\frac{xyz}{5x^{4}y^{3}z^{2}}=\frac{4x^{3}xy^{5}yz^{12}z}{5x^{4}y^{3}z^{2}}=\frac{4x^{4}y^{6}z^{13}}{5x^{4}y^{3}z^{2}}$

Since $x^{4}$ is in the numerator and the denominator, you can cancel it out.

$\small \frac{4x^{4}y^{6}z^{13}}{5x^{4}y^{3}z^{2}}=\frac{4y^{6}z^{13}}{5y^{3}z^{2}}$

Use the exponent rule $\frac{x^{a}}{x^{b}}=x^{a-b}$ to further simplify the expression by combining the terms $y^{6}$ and $y^{3}$ and $z^{13}$ and $\small z^{2}$ .

$\small \frac{4y^{6}z^{13}}{5y^{3}z^{2}}=\frac{4y^{3}z^{11}}{5}$

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Example Question #2 : How To Multiply Monomial Quotients

Simplify:

$\frac{a^{2}b^{4}}{4a^{5}}*\frac{2by^{4}}{y^{3}}$

Possible Answers:

$\frac{2b^{5}y}{4a^{3}}$

$\frac{2a^{3}}{yb^{5}}$

$\frac{yb^{5}}{2a^{3}}$

$\frac{a^{\frac{2}{5}}b^{5}}{2y^{\frac{4}{3}}}$

$\frac{2a^{2}b^{5}y^{4}}{4a^{5}y^{3}}$

Correct answer:

$\frac{yb^{5}}{2a^{3}}$

Explanation:

First, organize the variables so like terms are together in the numerator and in the denominator.

$\frac{2a^{2}b^{4}by^{4}}{4a^{5}y^{3}}$

Second, use rules of exponents to combine the following terms: $b^{4}$ and , $a^{2}$ and $a^{5}$ , and $y^{4}$ and $y^{3}$ . Remember the following exponent rules:

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

$\frac{x^{a}}{x^{b}}=x^{a-b}$

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

$\frac{2a^{2}b^{4}by^{4}}{4a^{5}y^{3}}=\frac{2a^{-3}b^{5}y}{4}=\frac{2b^{5}y}{4a^{3}}$

Third, divide the 2 and the 4 by the GCF, 2.

$\frac{2a^{2}b^{4}by^{4}}{4a^{5}y^{3}}=\frac{2b^{5}y}{4a^{3}}=\frac{yb^{5}}{2a^{3}}$

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Example Question #3 : How To Multiply Monomial Quotients

Simplify:

$\frac{24x^{5}}{6y^{2}}*\frac{14y^{4}x}{21}$

Possible Answers:

$y^{2}x^{6}$

$\frac{8y^{-2}}{3x^{5}}$

$\frac{8y^{2}x^{6}}{3}$

$\frac{8y^{4}x^{6}}{3y^{2}}$

$\frac{3}{8y^{2}x^{6}}$

Correct answer:

$\frac{8y^{2}x^{6}}{3}$

Explanation:

Divide 6 and 24 by 6 (the GCF) and 14 and 21 by 7 (the GCF). Combine like terms in the numerator and the denominator.

$\frac{24x^{5}}{6y^{2}}*\frac{14y^{4}x}{21}=\frac{4x^{5}}{y^{2}}*\frac{2y^{4}x}{3}=\frac{4*2*y^{4}x^{5}x}{3y^{2}}=\frac{8y^{4}x^{5}x}{3y^{2}}$

Use rules of exponents to combine the following terms: $y^{4}$ and $y^{2}$ and $x^{5}$ and Remember the following exponent rules:

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

and

$\frac{x^{a}}{x^{b}}=x^{a-b}$

$\frac{8y^{4}x^{5}x}{3y^{2}}=\frac{8y^{4}x^{6}}{3y^{2}}=\frac{8y^{2}x^{6}}{3}$

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Example Question #1 : How To Multiply Monomial Quotients

Simplify:

$\frac{10x^{3}}{b^{6}}*\frac{2b^{2}}{5x^{3}}$

Possible Answers:

$\frac{4x^{3}b^{2}}{x^{3}b^{6}}$

$\frac{20x^{3}b^{2}}{5x^{3}b^{6}}$

$\frac{b^{4}}{4}$

$\frac{4x^{3}}{x^{3}b^{4}}$

$\frac{4}{b^{4}}$

Correct answer:

$\frac{4}{b^{4}}$

Explanation:

Combine like terms in the numerator and the denominator. Then, divide 20 and 5 by 5 (the GCF).

$\frac{10 * 2*x^{3}b^{2}}{5x^{3}b^{6}}=\frac{20x^{3}b^{2}}{5x^{3}b^{6}}=\frac{4x^{3}b^{2}}{x^{3}b^{6}}$

The GCF rule can also be used to remove $x^{3}$ from the numerator and the deonominator. $x^{3}$ goes into $x^{3}$ once.

$\frac{4x^{3}b^{2}}{x^{3}b^{6}}=\frac{4b^{2}}{b^{6}}$

Use rules of exponents to combine the terms $b^{2}$ and $b^{6}$ . Remember the following exponent rules:

$\frac{x^{a}}{x^{b}}=x^{a-b}$

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

$\frac{4b^{2}}{b^{6}}=4b^{-4}=\frac{4}{b^{4}}$

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Example Question #4 : How To Multiply Monomial Quotients

Simplify:

$\frac{6x^{2}y^{2}z}{8z^{4}}*\frac{4xy^{3}}{20y}$

Possible Answers:

$\frac{3x^{3}y^{4}z^{5}}{20}$

$\small \frac{3x^{3}y^{4}}{20z^{3}}$

$\frac{3}{20}x^{3}y^{4}z^{4}$

$\frac{3x^{3}y^{6}z^{5}}{20}$

$\frac{3x^{3}y^{\frac{5}{1}}z^{\frac{1}{4}}}{20}$

Correct answer:

$\small \frac{3x^{3}y^{4}}{20z^{3}}$

Explanation:

In the first quotient, divide 6 and 8 by the GCF (2). In the second quotient, divide 4 and 20 by the GCF (4).

$\frac{6x^{2}y^{2}z}{8z^{4}}*\frac{4xy^{3}}{20y}=\frac{3x^{2}y^{2}z}{4z^{4}}*\frac{xy^{3}}{5y}$

Organize the variables so like terms are together in the numerator and in the denominator. Then use rules of exponents to combine $x^{2}$ and and $y^{2}$ and $y^{3}$ in the numerator. Remember the following exponent rule:

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

$\small \small \frac{3x^{2}y^{2}z}{4z^{4}}*\frac{xy^{3}}{5y}=\frac{3x^{2}xy^{2}y^{3}z}{20yz^{4}}=\frac{3x^{3}y^{5}z}{20yz^{4}}$

Use the rules of exponents

$\frac{x^{a}}{x^{b}}=x^{a-b}$

and

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

to further simplify the expression by combining the terms $\small y^{5}$ and $\small y$ , and $\small z$ and $\small z^{4}$ .

$\small \frac{3x^{3}y^{5}z}{20yz^{4}}=\frac{3x^{3}y^{4}z^{-3}}{20}=\frac{3x^{3}y^{4}}{20z^{3}}$

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Algebra 1 : Algebra 1

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #522 : Variables

Example Question #107 : Monomials

Example Question #1 : How To Multiply Monomial Quotients

Example Question #531 : Variables

Example Question #532 : Variables

Example Question #533 : Variables

Example Question #2 : How To Multiply Monomial Quotients

Example Question #3 : How To Multiply Monomial Quotients

Example Question #1 : How To Multiply Monomial Quotients

Example Question #4 : How To Multiply Monomial Quotients

All Algebra 1 Resources

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