Monomials - Algebra 1

Card 0 of 20

Question

Simplify the following:

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

Answer

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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Question

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

Answer

$p^{7}$ 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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Question

Simplify this expression:

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

Answer

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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Question

Simplify the following:

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

Answer

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

First, let us factor the numerator:

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

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Question

Simplify the rational expression.

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

Answer

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

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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Question

Divide the fractions.

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

Answer

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

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.

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

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Question

Simplify the fraction to its lowest terms:

$\frac{18x^4y^3z}{6xy^2z^3}$

Answer

The first step is to divide the constants, 18 and 6, by the LCM, 6, to get 3. When dividing variables, if the variable is present in the numerator and denominator, subtract the exponent found in the numerator by the exponent in the denominator.

For , you have .

For , you have .

For , you have .

Then write the simplified answer as one term:

$3x^3yz^{-2}$

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Question

Simplify the expression.

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

Answer

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

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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Question

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

Answer

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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Question

Evaluate

$\frac{(12w^{2}-15w+21w^{3})}{3w}$

Answer

$\frac{(12w^{2}-15w+21w^{3})}{3w}$

When dividing a polynomial by a monomial, we can use a divison called, term-by-term, dividing each of the top terms by the monomial.

$\frac{12w^{2}}{3w}-\frac{15w}{3w}+\frac{21w^{3}}{3w}$

Simplify.

$4w-5+7w^{2}$

Rewrite it with the leading coefficent first,

Final Answer: $7w^{2}+4w-5$

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Question

Divide:

$\frac{9x^4+6x^3-12x}{3x}$

Answer

$\frac{9x^4+6x^3-12x}{3x}=\frac{9x^4}{3x}+\frac{6x^3}{3x}-\frac{12x}{3x}=3x^3+2x^2-4$

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Question

Siimplify:

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

Answer

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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Question

Simplify:

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

Answer

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 .

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Question

Evaluate: $3x^3 \div 6x$

Answer

First, divide the coefficients.

$3\div 6 = \frac{1}{2}$ .

Now consider the variables. means x times itself 3 times. Doing $x^3 \div x$ is the same as $\frac{xxx}{x}$ .

We see that an x from the top can cancel with the x on the bottom, leaving $\frac{x*x}{1}$ , or . Our answer then is $\frac{1}{2}x^2$ .

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Question

Find the result of

$\frac{12x^3\cdot2x^4}{3x^5}$ .

Answer

The question asks for the quotient of $\frac{12x^3\cdot2x^4}{3x^5}$ .

We can rewrite the expression using the rules of multiplication and exponents to find the answer of .

$\frac{12\cdot2\cdot x\cdotx^{3+4}}{3x^5}= \frac{24}{3}\cdot x^{3+4-5}= 8x^2$

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Question

Divide: $\frac{x^2}{x^{-1}}$

Answer

Dividing exponents by the same base allows the powers to be subtracted. Rewrite the exponents as one base.

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

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Question

Simplify the following expression:

Which of the following expressions is equivalent to this one:

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

Answer

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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Question

Divide: $\frac{x}{x^3}$

Answer

Notice that both bases are the same. The power of the base in the numerator is one.

Since the bases are divided, simply subtract the powers.

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

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Question

Divide the following monomial quotients:

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

Answer

To solve this problem, split it into two steps:

1. Divide the coefficients

$\frac{6}{2}=3$

2. Divide the variables. We also need to remember the following laws of exponents rule: When dividing variables, subtract the exponents.

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

Combine these to get the final answer:

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Question

Divide the following monomial quotients:

$\frac{10x^{4}y}{5xy}$

Answer

To solve this problem, split it into two steps:

1. Divide the coefficients

$\frac{10}{5}=2$

2. Divide the variables. We also need to remember the following laws of exponents rule: When dividing variables, subtract the exponents.

$\frac{x^{4}}{x} =x^{4-1}=x^{3}$

$\frac{y}{y}=1$

Combine these to get the final answer:

$2x^{3}$

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