How to multiply monomial quotients - Algebra 1

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Question

Multiply, expressing the product in simplest form:

$\small \frac{15x^{5}}{14y^{3}} \cdot \frac{21y^{2} }{25x^{3}}$

Answer

Cross-cancel the coefficients by dividing both 15 and 25 by 5, and both 14 and 21 by 7:

$\small \small \small \small \frac{15x^{5}}{14y^{3}} \cdot \frac{21y^{2} }{25x^{3}} = \small \small \small \small \frac{3x^{5}}{2y^{3}} \cdot \frac{3y^{2} }{5x^{3}} = \frac{3x^{5} \cdot 3y^{2}}{5x^{3} \cdot 2y^{3}} = \frac{3 \cdot 3 \cdot x^{5} y^{2}}{5\cdot 2\cdot x^{3} y^{3}} = \frac{9 x^{5} y^{2}}{10 x^{3} y^{3}}$

Now use the quotient rule on the variables by subtracting exponents:

$\small \frac{9 x^{5} y^{2}}{10 x^{3} y^{3}} = \frac{9 x^{5-3}}{10 y^{3-2}} = \frac{9 x^{2}}{10 y^{1}} = \frac{9 x^{2}}{10 y}$

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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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.

$\frac{24x^{5}}{6y^{2}}\frac{14y^{4}x}{21}=\frac{4x^{5}}{y^{2}}\frac{2y^{4}x}{3}=\frac{42y^{4}x^{5}x}{3y^{2}}=\frac{8y^{4}x^{5}x}{3y^{2}}$

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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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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.

$\frac{3a^{3}bc}{a^{5}b^{7}}\frac{5b^{4}c^{2}}{4a}=\frac{35a^{3}bb^{4}cc^{2}}{4a^{5}ab^{7}}$

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: $x^{a}x^{b}=x^{a+b}$ ,

$\frac{35a^{3}bb^{4}cc^{2}}{4a^{5}ab^{7}}=\frac{15a^{3}b^{5}c^{3}}{4a^{6}b^{7}}$

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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Question

Simplify the following:

$\frac{2xy^2}{3yz^2} \times \frac{15y^2z}{14x^2y}$

Answer

In this problem, you have two fractions being multiplied. You can first simplify the coefficients in the numerators and denominators. You can divide and cancel the 2 and 14 each by 2, and the 3 and 15 each by 3:

$\frac{2xy^2}{3yz^2} \times \frac{15y^2z}{14x^2y} = \frac{xy^2}{yz^2} \times \frac{5y^2z}{7x^2y}$

You can multiply the two numerators and two denominators, keeping in mind that when multiplying like variables with exponents, you simplify by adding the exponents together:

$\frac{xy^2}{yz^2} \times \frac{5y^2z}{7x^2y} = \frac{5xy^4z}{7x^2y^2z^2}$

Any variables that are both in the numerator and denominator can be simplified by subtracting the numerator's exponent by the denominator's exponent. If you end up with a negative exponent in the numerator, you can move the variable to the denominator to keep the exponent positive:

$\frac{5xy^4z}{7x^2y^2z^2} = \frac{5y^2}{7xz}$

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Question

Simplify:

$\frac{9ab^2c}{4c^3} \times \frac{a^2b^2}{3a^3c}$

Answer

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators, by dividing 9 and 3 each by 3:

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

Next you can multiply the two numerators, and multiply the two denominators. Remember that when multiplying like variables with exponents, you add the exponents together:

$\frac{3ab^2c}{4c^3} \times \frac{a^2b^2}{a^3c} = \frac{3a^3b^4c}{4a^3c^4}$

If a variable shows up in both the numerator and denominator, you can simplify by subtracting the numerator's exponent by the denominator's exponent. If you end up with a negative exponent in the numerator, you can move the variable and exponent to the denominator to make it positive:

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

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Question

Simplify the following:

$\frac{10x^2y^3z}{7xz^3} \times \frac{21xyz^2}{5y}$

Answer

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators, by dividing 10 and 5 each by 5 and dividing 21 and 7 each by 7:

$\frac{10x^2y^3z}{7xz^3} \times \frac{21xyz^2}{5y} = \frac{2x^2y^3z}{xz^3} \times \frac{3xyz^2}{y}$

Next you can multiply the two numerators, and multiply the two denominators. Remember that when multiplying like variables with exponents, you add the exponents together:

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

If a variable shows up in both the numerator and denominator, you can simplify by subtracting the numerator's exponent by the denominator's exponent:

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

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Question

Simplify the following:

$\frac{2a^3bc}{3a^2bc^4} \times \frac{15ab^3c^2}{5a^3b^5c^2}$

Answer

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators. The two coefficients in the denominators multiply up to 15, allowing you to divide and cancel those two coefficients with the 15 in the numerator:

$\frac{2a^3bc}{3a^2bc^4} \times \frac{15ab^3c^2}{5a^3b^5c^2} = \frac{2a^3bc}{a^2bc^4} \times \frac{ab^3c^2}{a^3b^5c^2}$

Next you can multiply the two numerators, and multiply the two denominators. Remember that when multiplying like variables with exponents, you add the exponents together:

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

If a variable shows up in both the numerator and denominator, you can simplify by subtracting the numerator's exponent by the denominator's exponent. If you end up with a negative exponent in the numerator, you can move the variable and exponent to the denominator to make it positive:

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

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Question

Multiply. Give the answer in simplest form.

$\left(\frac{3}{2x^2} \right )\cdot\left(\frac{4x^2y}{9} \right )$

Answer

Multiplying quotients is similar to multiplying fractions, so we multiply straight across to get $\small \frac{12x^2y}{18x^2}$ . From this point, we can simplify. Since $\small \frac{12}{18}=\frac{2}{3}$ and $\small \frac{x^2}{x^2}=x^{2-2}=x^0=1$ , our final answer becomes $\small \frac{2}{3} \cdot 1 \cdot y = \frac{2y}{3}$ .

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Question

Simplify:

$\frac{30a^{2}b}{c}*\frac{bc^{2}}{5c}$

Answer

Combine like terms in the numerator and the denominator. Use the rules of exponents to combine and in the numerator and and in the denominator. Remember that $x^{a}x^{b}=x^{a+b}$ Then, divide 30 by 5 (the GCF).

$\frac{30a^{2}bbc^{2}}{5cc}=\frac{30a^{2}b^{2}c^{2}}{5c^{2}}=\frac{6a^{2}b^{2}c^{2}}{c^{2}}$

The GCF rule can also be used to remove $c^{2}$ from the numerator and the deonominator. $c^{2}$ goes into $c^{2}$ once.

$\frac{6a^{2}b^{2}c^{2}}{c^{2}}=6a^{2}b^{2}$

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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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: $x^{a}x^{b}=x^{a+b}$ ,

$\frac{7a^{7}b^{6}b^{3}c^{4}}{83a^{3}b^{2}}=\frac{7a^{7}b^{6}b^{3}c^{4}}{24a^{3}b^{2}}=\frac{7a^{7}b^{9}c^{4}}{24a^{3}b^{2}}$

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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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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. You cannot further combine the variables because each variable is represented by a different letter. You cannot further reduce the integers because they do not have a common factor.

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

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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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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Question

Simplify:

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

Answer

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.

$\frac{3a^{2}}{b^{4}}\frac{4b}{a}\frac{a^{2}b^{4}}{2}=\frac{3a^{2}}{b^{4}}\frac{2b}{a}\frac{a^{2}b^{4}}{1}=\frac{23a^{2}a^{2}bb^{4}}{ab^{4}}=\frac{6a^{2}a^{2}bb^{4}}{ab^{4}}$

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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Question

Multiply the following monomial quotients:

$5x\cdot2x^{2}$

Answer

To solve this problem, split it into two steps:

1. Multiply the coefficients

$5\cdot2=10$

2. Multiply the variables. We also need to remember the following laws of exponents rule: When multiplying variables, add the exponents.

$x\cdot x^{2}=x^{1+2}=x^{3}$

Combine these to get the final answer:

$10x^{3}$

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Question

Multiply the following monomial quotients:

$6\cdot3x$

Answer

To solve this problem, split it into two steps:

1. Multiply the coefficients

$6\cdot3=18$

2. Combine this number with the single variable to get the final answer:

$18\cdot x=18x$

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