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Algebra 1 : How to multiply monomial quotients

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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:

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

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

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

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

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

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 #2 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

$4x^{6}yz$

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

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

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

$4x^{6}z$

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 #3 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

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

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

$6a^{3}b$

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

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

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 #4 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #5 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #6 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #7 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #1 : How To Multiply Monomial Quotients

Simplify:

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

Possible Answers:

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

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

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

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

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

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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Example Question #1 : How To Multiply Monomial Quotients

Simplify:

$\frac{27a^{3}bc}{9a^{5}b^{7}}*\frac{20b^{4}c^{2}}{16a}$

Possible Answers:

$\frac{15}{4a^{3}b^{2}c^{3}}$

$\frac{4a^{3}b^{2}}{15c^{3}}$

$\frac{15a^{3}b^{2}c^{3}}{4}$

$\frac{15c^{3}}{4a^{3}b^{2}}$

$\frac{15a^{\frac{1}{2}}b^{\frac{5}{7}}c^{3}}{4}$

Correct answer:

$\frac{15c^{3}}{4a^{3}b^{2}}$

Explanation:

First, in the first quotient, divide 27 and 9 by the GCF (9). In the second quotient, divide 20 and 16 by the GCF (4).

$\frac{27a^{3}bc}{9a^{5}b^{7}}*\frac{20b^{4}c^{2}}{16a}=\frac{3a^{3}bc}{a^{5}b^{7}}*\frac{5b^{4}c^{2}}{4a}$

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

$\frac{3a^{3}bc}{a^{5}b^{7}}*\frac{5b^{4}c^{2}}{4a}=\frac{3*5*a^{3}bb^{4}cc^{2}}{4a^{5}ab^{7}}$

Third, multiply the integers and use the rules of exponents to combine the following terms: $a^{5}$ and , and $b^{4}$ , and and $c^{2}$ . Remember the following exponent rule: $x^{a}*x^{b}=x^{a+b}$ ,

$\frac{3*5*a^{3}bb^{4}cc^{2}}{4a^{5}ab^{7}}=\frac{15a^{3}b^{5}c^{3}}{4a^{6}b^{7}}$

Fourth, 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 $a^{3}$ and $a^{6}$ and $b^{5}$ and $b^{7}$ .

$\frac{15a^{3}b^{5}c^{3}}{4a^{6}b^{7}}=\frac{15a^{-3}b^{-2}c^{2}}{4}=\frac{15c^{3}}{4a^{3}b^{2}}$

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Example Question #10 : How To Multiply Monomial Quotients

Simplify:

$\frac{a^{7}b^{6}c^{4}}{8}*\frac{28b^{3}}{12a^{3}b^{2}}$

Possible Answers:

$\frac{24}{7a^{4}b^{7}c^{4}}$

$\frac{7c^{4}}{6a^{10}b^{11}}$

$\frac{7a^{4}b^{7}c^{4}}{24}$

$\frac{7a^{4}b^{7}c^{4}}{6}$

$\frac{7a^{\frac{7}{3}}b^{\frac{9}{2}}c^{4}}{24}$

Correct answer:

$\frac{7a^{4}b^{7}c^{4}}{24}$

Explanation:

Because 28 and 12, the coefficients in the second quotient, share common factors, you can divide them by the GCF (4). Because the cofficients in the first quotient are technically 1 and 8, you cannot further reduce the 8.

$\frac{a^{7}b^{6}c^{4}}{8}*\frac{28b^{3}}{12a^{3}b^{2}}=\frac{a^{7}b^{6}c^{4}}{8}*\frac{7b^{3}}{3a^{3}b^{2}}$

Organize the variables so like terms are together in the numerator and in the denominator. Then, multiply and in the denominator and use rules of exponents to combine $b^{6}$ and $b^{3}$ in the numerator. Remember the following exponent rule: $x^{a}*x^{b}=x^{a+b}$ ,

$\frac{7a^{7}b^{6}b^{3}c^{4}}{8*3a^{3}b^{2}}=\frac{7a^{7}b^{6}b^{3}c^{4}}{24a^{3}b^{2}}=\frac{7a^{7}b^{9}c^{4}}{24a^{3}b^{2}}$

Use the exponent rule

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

to further simplify the expression by combining the terms $a^{7}$ and $a^{3}$ and $b^{9}$ and $b^{2}$ .

$\frac{7a^{7}b^{9}c^{4}}{24a^{3}b^{2}}=\frac{7a^{4}b^{7}c^{4}}{24}$

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Algebra 1 : How to multiply monomial quotients

Study concepts, example questions & explanations for Algebra 1

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Example Question #1 : How To Multiply Monomial Quotients

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