Monomials

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Algebra › Monomials

Questions 1 - 10

Simplify the expression.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

$\small \frac{xy^3}{z^3}$

$\small \frac{x^2y^5}{z^3}$

$\small \frac{z^3}{x^2y^5}$

$\small \frac{z^3}{xy^3}$

Explanation

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}=\frac{x^2y^2x^3y^4z}{x^4y^3z^4}$

To combine like terms in the numerator, we add their exponents.

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

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

$x^{5-4}y^{6-3}z^{1-4}=xy^3z^{-3}$

Remember that any negative exponents stay in the denominator.

$\frac{xy^3}{z^3}$

Simplify the expression.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

$\small \frac{xy^3}{z^3}$

$\small \frac{x^2y^5}{z^3}$

$\small \frac{z^3}{x^2y^5}$

$\small \frac{z^3}{xy^3}$

Explanation

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}=\frac{x^2y^2x^3y^4z}{x^4y^3z^4}$

To combine like terms in the numerator, we add their exponents.

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

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

$x^{5-4}y^{6-3}z^{1-4}=xy^3z^{-3}$

Remember that any negative exponents stay in the denominator.

$\frac{xy^3}{z^3}$

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 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 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:

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

$4x^{6}z$

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

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

$4x^{6}yz$

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

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:

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

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

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

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

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

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

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).

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

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

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

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).

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

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

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:

$5 \times 10xy$

$50x \times 50y$

$15x \times y$

The answers provided do not show the correct simplificaiton.

Explanation

When multiplying a whole number by a polynomial, we simply multiply that number by whatever coefficient is present in front of the variables of the polynomial. We then maintain the variables in the simplified expression.

$10 \times 5 = 50$