Algebra - How to multiply monomial quotients

Simplify the following:

$\frac{x}{y}\cdot\frac{y}{z}$

$\frac{x}{z}$

$\frac{z}{x}$

$\frac{y^2}{xz}$

$\frac{xz}{y^2}$

Explanation

To solve, simply multiply numerators and denominators and then simplify. Thus,

$\frac{x}{y}\cdot\frac{y}{z}=\frac{xy}{yz}=\frac{x}{z}$

Simplify:

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

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

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

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

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

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

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 $\small z^{4}$ .

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

Simplify:

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

$4x^{6}z$

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

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

$4x^{6}yz$

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

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 .

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

Simplify:

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

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

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

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

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

Explanation

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:

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

Multiply the following monomial quotients:

$6x\cdot 7x^{2}$

$42x^{3}$

$13x^{3}$

$42x^{2}$

Explanation

To solve this problem, split it into two steps:

1. Multiply the coefficients

$6\cdot7=42$

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:

$42x^{3}$

Multiply the following monomial quotients:

$2x^{3}\cdot x^{5}=2x^{8}$

$2x^{8}$

$3x^{8}$

$2x^{15}$

$4x^{12}$

Explanation

To solve this problem, split it into two steps:

1. Multiply the coefficients

$2\cdot1=2$

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

$x^{3}\cdot x^{5}=x^{3+5}=x^{8}$

Combine these to get the final answer:

$2x^{8}$

Multiply the following monomial quotients:

$2x^{5}\cdot 3x^{2}$

$6x^{7}$

$5x^{7}$

$6x^{3}$

$6x^{10}$

Explanation

To solve this problem, split it into two steps:

1. Multiply the coefficients

$2\cdot3=6$

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

$x^{5}\cdot x^{2}=x^{5+2}=x^{7}$

Combine these to get the final answer:

$6x^{7}$

Multiply the following monomial quotients:

$3x\cdot 2x$

$6x^{2}$

$4x^{2}$

Explanation

To solve this problem, split it into two steps:

1. Multiply the coefficients

$3\cdot2=6$

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

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

Combine these to get the final answer:

$6x^{2}$

Multiply the following monomial quotients:

$12x^{2}\cdot 4x^{3}$

$48x^{5}$

$16x^{5}$

$48x^{6}$

Explanation

To solve this problem, split it into two steps:

1. Multiply the coefficients

$12\cdot4=48$

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

$x^{2}\cdot x^{3}=x^{2+3}=x^{5}$

Combine these to get the final answer:

$48x^{5}$

Simplify:

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

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

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

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

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

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

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 $c^{4}$ .