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Algebra 1 : Variables

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Example Questions

Example Question #1 : How To Divide Monomial Quotients

Siimplify:

$\frac{2x^{2}y^{3}z - 6xy^{2}z^{2}+10x^{3}yz^{3}}{2xyz} =\ ?$

Possible Answers:

$xy - yz+5xz^{2}$

$xy^{2}-3yz+5x^{2}z^{2}$

$xy^{2}-3xz+5xy$

$x^{2}y - 3xyz + 5x^{2}yz^{2}$

$xyz-3yz+5x^{2}yz^{2}$

Correct answer:

$xy^{2}-3yz+5x^{2}z^{2}$

Explanation:

For any polynomial division, divide each term in the numerator individually by the denominator:

$\frac{2x^{2}y^{3}z - 6xy^{2}z^{2}+10x^{3}yz^{3}}{2xyz} =$ $\frac{2x^{2}y^{3}z}{2xyz} - \frac{6xy^{2}z^{2}}{2xyz} + \frac{10x^{3}yz^{3}}{2xyz}=$

$xy^{2}-3yz+5x^{2}z^{2}$

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Example Question #4 : How To Divide Monomial Quotients

Simplify:

$\frac{14x^2y}{7x}$

Possible Answers:

$\frac{1}{2xy}$

98x^3y

98xy

2xy

Correct answer:

2xy

Explanation:

When dividing monomials, consider the coefficients and variables separately. Rewrite the expression as $\left(\frac{14}{7}\right)\left(\frac{x^2}{x}\right)\left(y\right)$ , grouping common bases. For the coeffiecients, we can divide normally: $\left(\frac{14}{7} \right )=2$ . For the variables, we can keep the common base and subtract the exponents: $\left(\frac{x^2}{x} \right )=x^{2-1}=x$ . Then, multiply each portion all back together to obtain 2xy .

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Example Question #1 : Simplifying Expressions

Simplify the expression.

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

Possible Answers:

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

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

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

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

Correct answer:

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

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.

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

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Example Question #1 : Simplifying Expressions

Simplify: $\frac{5p^{7}}{25p^{^{4}}}$

Possible Answers:

$\frac{p^{7/4}}{5}$

$\frac{p}{5}$

$\frac{p^{3}}{5}$

$\frac{p^{-3}}{5}$

$5p^{3}$

Correct answer:

$\frac{p^{3}}{5}$

Explanation:

$p^{7}$ and $p^{4}$ cancel out, leaving $p^{3}$ in the numerator. 5 and 25 cancel out, leaving 5 in the denominator

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Example Question #1 : Simplifying Expressions

Simplify the following:

$\frac{4x+6x^2}{2x}$

Possible Answers:

2x(2-3x)

2+3x

$\frac{2-3x}{2x}$

$\frac{2}{2x}-\frac{3}{2}$

$\frac{1}{x}-1\frac{1}{2}$

Correct answer:

2+3x

Explanation:

$\frac{4x+6x^2}{2x}$

First, let us factor the numerator:

4x+6x^2=x(4+6x)=2x(2+3x)

$\frac{2x(2+3x)}{2x}=2+3x$

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Example Question #2 : How To Divide Monomial Quotients

Simplify the following:

$\frac{-2pl^2}{3yz}\div \frac{9lp^2}{5xy}$

Possible Answers:

$\frac{-18}{15xyz}$

$\frac{5}{9}$

$\frac{-18p^{3}l^{3}}{15xy^{2}z}$

$\frac{-10lx}{27zp}$

Correct answer:

$\frac{-10lx}{27zp}$

Explanation:

First, flip the numerator and the denominator of the second fraction to turn the division into multiplication.

$\frac{-2pl^2}{3yz}\div \frac{9lp^2}{5xy}=\frac{-2pl^2}{3yz} * \frac{5xy}{9lp^2}=\frac{-2pl^25xy}{3yz9lp^2}$

We can then cancel like terms.

From both the numerator and denominator, remove one , remove one , and remove one :

$\frac{-2l5x}{3z9p}$

Then we finish by multiplying the constants:

$\frac{-10lx}{27zp}$

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Example Question #1 : How To Divide Monomial Quotients

Simplify this expression:

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

Possible Answers:

$xyz^{2}$

$\frac{yz^{2}}{x}$

$\frac{z^{2}}{xy}$

$\frac{1}{xyz^{2}}$

$\frac{xy}{z^{2}}$

Correct answer:

$\frac{z^{2}}{xy}$

Explanation:

When different powers of the same variable are multiplied, the exponents are added. When different powers of the same variable are divided, the exponents are subtracted. So, as an example:

$\frac{x^{3}}{x} = x^{\left ( 3-1\right )} = x^{2}$

For the above problem,

$\frac{x^{2}}{x^{3}} = \frac{1}{x} ; \frac{y^{-3}}{y^{-2}} = \frac{y^{2}}{y^{3}} = \frac{1}{y} ; \frac{z^{4}}{z^{2}} = z^{2}$

Therefore, the expression simplifies to:

$\left ( \frac{1}{x} \right ) \left ( \frac{1}{y} \right ) z^{2} = \frac{z^{2}}{xy}$

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Example Question #472 : Variables

Simplify the rational expression.

$\frac{12x^2y^4z^6}{6xz^{12}}$

Possible Answers:

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

$\frac{x}{2z^2}$

$\frac{2xy^4}{z^2}$

$\frac{2xy^4}{z^6}$

Correct answer:

$\frac{2xy^4}{z^6}$

Explanation:

$\frac{12x^2y^4z^6}{6xz^{12}}$

To simplify variables with exponents through division, you must subtract the exponent in the denominator from the numerator.

$\frac{12}{6}(x^{2-1})(y^{4-0})(z^{6-12})=2xy^4z^{-6}=\frac{2xy^4}{z^6}$

Remember that negative exponents will eventually be moved back to the denominator.

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Example Question #123 : Basic Single Variable Algebra

Divide the fractions.

$\frac{30xy}{11} \div \frac{25x^{3}}{22y}$

Possible Answers:

$\frac{12y^{2}}{5x^{2}}$

$\frac{5x^{4}}{12}$

$\frac{5y^{2}}{12x^{2}}$

$\frac{5x^{2}}{12y^{2}}$

$\frac{12x^{4}}{5}$

Correct answer:

$\frac{12y^{2}}{5x^{2}}$

Explanation:

$\frac{30xy}{11} \div \frac{25x^{3}}{22y}$

Dividing fractions is equal to multiplying by the reciprocal.

$\frac{30xy}{11} * \frac{22y}{25x^{3}}$

Cross-cancel the 30 with the 25, and the 22 with the 11.

$\frac{6xy}{1} *\frac{2y}{5x^{3}}$

Combine via multiplication and simplify.

$\frac{6xy* 2y}{5x^{3}}=\frac{6* 2*x* y*y}{5x^{3}}=\frac{12y^{2}}{5x^{2}}$

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Example Question #473 : Variables

Simplify $\frac{a^{4}b^{2}}{6a^{3}b}\div \frac{2b^{5}}{12a^{7}}$

Possible Answers:

$\frac{b^4}{a^8}$

$\frac{a^8}{b^4}$

$\frac{b^6}{36a^6}$

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

Correct answer:

$\frac{a^8}{b^4}$

Explanation:

Rewrite so that you are multiplying the reciprocal of the second fraction:

$\frac{a^{4}b^{2}}{6a^{3}b}\times \frac{12a^{7}}{2b^{5}}$

You can then simplify using rules of exponents:

$\frac{12a^{11}b^2}{12a^3b^6}=\frac{a^8}{b^4}$

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Algebra 1 : Variables

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Divide Monomial Quotients

Example Question #4 : How To Divide Monomial Quotients

Example Question #1 : Simplifying Expressions

Example Question #1 : Simplifying Expressions

Example Question #1 : Simplifying Expressions

Example Question #2 : How To Divide Monomial Quotients

Example Question #1 : How To Divide Monomial Quotients

Example Question #472 : Variables

Example Question #123 : Basic Single Variable Algebra

Example Question #473 : Variables

All Algebra 1 Resources

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