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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #3 : Simplifying Exponents

Simplify:

$\frac{x^{6}zy^{-9}}{x^{-4}z^2y^3}$

Possible Answers:

$\frac{x^{10}z^3}{y^6}$

$x^2z^{-1}y^6$

$\frac{y^6}{x^{10}z^3}$

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

$\frac{x^{10}}{zy^{12}}$

Correct answer:

$\frac{x^{10}}{zy^{12}}$

Explanation:

Step 1: Use the division of exponents rule. Subtract the exponents in the numerator from the exponents in the denominator.

${x^{6+4}z^{1-2}y^{-9-3}$

$x^{10}z^{-1}y^{-12}$

Step 2: Represent the negative exponents as positive ones by moving them to the denominator:

${\color{Red} \frac{x^{10}}{zy^{12}}}$

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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 #11 : Multiplying And Dividing Exponents

Simplify:

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

Possible Answers:

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

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

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

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

Correct answer:

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

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:

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

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Example Question #12 : Multiplying And Dividing Exponents

Simplify the following:

$\frac{10x^2y^3z}{7xz^3} \times \frac{21xyz^2}{5y}$

Possible Answers:

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

6x^2y^3z

6x^2y^3

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

Correct answer:

6x^2y^3

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 10 and 5 each by 5 and dividing 21 and 7 each by 7:

$\frac{10x^2y^3z}{7xz^3} \times \frac{21xyz^2}{5y} = \frac{2x^2y^3z}{xz^3} \times \frac{3xyz^2}{y}$

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:

$\frac{2x^2y^3z}{xz^3} \times \frac{3xyz^2}{y} = \frac{6x^3y^4z^3}{xyz^3}$

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:

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

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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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Example Question #3471 : Algebra Ii

Simplify the following:

$\frac{2a^3bc}{3a^2bc^4} \times \frac{15ab^3c^2}{5a^3b^5c^2}$

Possible Answers:

$\frac{2}{ab^2c^3}$

$\frac{2ab}{c^3}$

$\frac{2c^3}{ab^2}$

2ab^2c^3

Correct answer:

$\frac{2}{ab^2c^3}$

Explanation:

In this problem, you have two fractions being multiplied. You can first divide and cancel the coefficients in the numerators and denominators. The two coefficients in the denominators multiply up to 15, allowing you to divide and cancel those two coefficients with the 15 in the numerator:

$\frac{2a^3bc}{3a^2bc^4} \times \frac{15ab^3c^2}{5a^3b^5c^2} = \frac{2a^3bc}{a^2bc^4} \times \frac{ab^3c^2}{a^3b^5c^2}$

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:

$\frac{2a^3bc}{a^2bc^4} \times \frac{ab^3c^2}{a^3b^5c^2} = \frac{2a^4b^4c^3}{a^5b^6c^6}$

$\frac{2a^4b^4c^3}{a^5b^6c^6} = \frac{2}{ab^2c^3}$

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Example Question #3472 : Algebra Ii

Simplify $x^3\cdot x^{-2}\cdot x^4$ .

Possible Answers:

$\frac{1}{x^5}$

$x^{-24}$

$x^{-12}$

x^5

$\frac{1}{x^{24}}$

Correct answer:

x^5

Explanation:

When multiplying variables with the same base, you add the exponents. So, $x^{3+(-2)+4}=x^{1+4}=x^5$ .

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Example Question #13 : Multiplying And Dividing Exponents

Simplify the expression.

y^2z^3y^4z^5y^6

Possible Answers:

$y^9z^{11}$

$y^{12}z^8$

$y^{15}z^3$

$y^{12}z^{10}$

$y^{10}z^8$

Correct answer:

$y^{12}z^8$

Explanation:

y^2z^3y^4z^5y^6

Rearrange the expression so that the $\small y$ and $\small z$ variables of different powers are right next to each other.

y^2y^4y^6z^3z^5

When multiplying the same variable with different exponents, it is the same as adding the exponents: $a^ba^c=a^{b+c}$ . Taking advantage of this rule, the problem can be rewritten.

$y^{2+4+6}z^{3+5}$

$y^{6+6}z^8$

$y^{12}z^8$

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Example Question #4775 : Algebra 1

Simplify:

$\frac{30a^{2}b}{c}*\frac{bc^{2}}{5c}$

Possible Answers:

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

$6a^{2}b^{2}$

$6ab^{2}$

$6a^{2}b^{2}c^{4}$

$\frac{6a^{2}b^{2}c^{2}}{c^{2}}$

Correct answer:

$6a^{2}b^{2}$

Explanation:

Combine like terms in the numerator and the denominator. Use the rules of exponents to combine and in the numerator and and in the denominator. Remember that $x^{a}*x^{b}=x^{a+b}$ Then, divide 30 by 5 (the GCF).

$\frac{30a^{2}b*b*c^{2}}{5c*c}=\frac{30a^{2}b^{2}c^{2}}{5c^{2}}=\frac{6a^{2}b^{2}c^{2}}{c^{2}}$

The GCF rule can also be used to remove $c^{2}$ from the numerator and the deonominator. $c^{2}$ goes into $c^{2}$ once.

$\frac{6a^{2}b^{2}c^{2}}{c^{2}}=6a^{2}b^{2}$

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3 : Simplifying Exponents

Example Question #1 : How To Divide Monomial Quotients

Example Question #472 : Variables

Example Question #11 : Multiplying And Dividing Exponents

Example Question #12 : Multiplying And Dividing Exponents

Example Question #473 : Variables

Example Question #3471 : Algebra Ii

Example Question #3472 : Algebra Ii

Example Question #13 : Multiplying And Dividing Exponents

Example Question #4775 : Algebra 1

All Algebra II Resources

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