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Algebra 1 : Monomials

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Example Questions

Example Question #524 : Variables

Divide: $\frac{18x^2y^3}{36x^5y^{-3}}$

Possible Answers:

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

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

$\frac{1}{2x^3}$

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

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

Correct answer:

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

Explanation:

When similar bases of a certain power are divided, their powers can be subtracted.

Evaluate each term.

$\frac{18}{36} = \frac{1}{2}$

When dealing with negative exponents, they can be rewritten as the reciprocal of the positive exponent.

$\frac{x^2}{x^5} = x^{2-5} = x^{-3} = \frac{1}{x^3}$

$\frac{y^3}{y^{-3}} = y^{3-(-3)} = y^6$

Multiply all the terms together.

$\frac{1}{2}\cdot\frac{1}{x^3}\cdot y^6 = \frac{y^6}{2x^3}$

The answer is: $\frac{y^6}{2x^3}$

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Example Question #525 : Variables

Divide the monomials: $\frac{3x^3y}{6xy^2z}$

Possible Answers:

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

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

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

$\frac{1}{2x^2yz}$

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

Correct answer:

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

Explanation:

Rewrite both the numerator and denominator by factors.

$\frac{3x^3y}{6xy^2z} = \frac{3 \times x \times x \times x \times y}{3 \times 2 \times y \times y \times z}$

Cancel all the common terms on the top and bottom.

The answer is: $\frac{x^2}{2yz}$

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Example Question #526 : Variables

Divide the monomials: $\frac{3x^2y^5z}{3x^2y^{-5}z^3}$

Possible Answers:

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

$\frac{y^{10}}{z^2}$

$\frac{1}{y^{10}z^2}$

$y^{10}z^2$

$\frac{z^2}{y^{10}}$

Correct answer:

$\frac{y^{10}}{z^2}$

Explanation:

Some of the terms of the given expression can already be simplified.

Cancel out the 3x^2 terms.

$\frac{y^5z}{y^{-5}z^3}$

When we are dividing powers of a similar base, it is the same as subtracting the exponents.

$\frac{y^5}{y^{-5}} = y^{5-(-5)}= y^{10}$

$\frac{z}{z^3} = z^{1-3} = z^{-2} = \frac{1}{z^2}$

A negative exponent can be written as one over the base's positive exponent.

Multiply both terms together.

$y^{10} \cdot \frac{1}{z^2}= \frac{y^{10}}{z^2}$

The answer is: $\frac{y^{10}}{z^2}$

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Example Question #51 : How To Divide Monomial Quotients

Simplify: $\frac{8xz^3}{2z}$

Possible Answers:

4z^2

4xz^2

4x^2z^2

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

4x-z^2

Correct answer:

4xz^2

Explanation:

In order to simplify this fraction, we will need to rewrite the numerator using factors.

$\frac{8xz^3}{2z}=\frac{2\cdot 4 \cdot x \cdot z^3}{2z}$

Cancel the common terms in the numerator and denominator.

The answer is: 4xz^2

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Example Question #61 : How To Divide Monomial Quotients

Divide the monomials: $\frac{3x^6yz^3}{3x^{-6}y^6z^5}$

Possible Answers:

y^5z^2

$\frac{1}{y^5z^2}$

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

$\frac{y^5}{z^2}$

Correct answer:

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

Explanation:

When powers of a similar base are divided, the powers can be subtracted.

The coefficients will cancel.

Subtract each power for each variable.

$x^{6-(-6)}=x^{12}$

$y^{1-6}= y^{-5} = \frac{1}{y^5}$

$z^{3-5}= z^{-2} = \frac{1}{z^2}$

Multiply and combine each of these terms together.

$x^{12} \times \frac{1}{y^5} \times \frac{1}{z^2}$

The answer is: $\frac{x^{12}}{y^5z^2}$

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Example Question #522 : Variables

Divide: $\frac{9ab^6}{36a^3b^6}$

Possible Answers:

$\frac{1}{27a^2}$

4a^2

$\textup{Cannot be solved.}$

$\frac{1}{4a^2}$

Correct answer:

$\frac{1}{4a^2}$

Explanation:

Rewrite the numerator and denominator in factors.

$\frac{9ab^6}{36a^3b^6} = \frac{9\times a \times b^6}{9 \times 4 \times a\times a^2 \times b^6}$

Cancel out the common terms in the numerator and denominator. Since all the terms in the numerator cancels out, the numerator becomes a one.

The answer is: $\frac{1}{4a^2}$

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Example Question #107 : Monomials

Simplify the following expression:

Which of the following expressions is equivalent to this one:

$\frac{4a^{3}b^{2}}{6a}\cdot\frac{8c^{4}b^{3}}{2b}$

Possible Answers:

$\frac{16a^2b^3c^4}{3}$

$\frac{8a^{2}b^{4}c^{4}}{3}$

$\frac{8a^{2}b^{3}c^{4}}{3}$

$\frac{32a^3b^3 c^4}{12}$

$\frac{8a^2bc^4}{6}$

Correct answer:

$\frac{8a^{2}b^{4}c^{4}}{3}$

Explanation:

First, simplify each term individually. Let's start with the term on the left. We can factor out from both the numerator and denominator.

$\frac{4a^{3}b^{2}}{6a}\rightarrow \frac{2a^2b^2}{3}$

Next, factor out from the numerator and denominator of the term on the right.

$\frac{8c^{4}b^{3}}{2b}\rightarrow \frac{4c^4b^2}{1}$

Combine and multiply. Remember that when we are multiplying terms with exponents, we need to add their exponential values. Likewise, when we divide exponents, we will subtract their exponential values.

$\frac{4a^3b^2}{6a}\cdot\frac{8c^4b^3}{2b} = \frac{2a^2b^2}{3}\cdot\frac{4c^4b^2}{1}$

$\frac{2a^2b^2}{3}\cdot\frac{4c^4b^2}{1}=\frac{(2a^2b^2)(4c^4b^2)}{3}$

Simplify.

$\frac{(2a^2b^2)(4c^4b^2)}{3}=\frac{8a^2b^4c^4}{3}$

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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:

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

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

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

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

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

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 #531 : Variables

Simplify:

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

Possible Answers:

$4x^{6}yz$

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

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

$4x^{6}z$

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

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 #532 : Variables

Simplify:

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

Possible Answers:

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

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

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

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

$6a^{3}b$

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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Algebra 1 : Monomials

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #524 : Variables

Example Question #525 : Variables

Example Question #526 : Variables

Example Question #51 : How To Divide Monomial Quotients

Example Question #61 : How To Divide Monomial Quotients

Example Question #522 : Variables

Example Question #107 : Monomials

Example Question #1 : How To Multiply Monomial Quotients

Example Question #531 : Variables

Example Question #532 : Variables

All Algebra 1 Resources

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