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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #35 : Simplifying Exponents

Simplify the following:

$\frac{x^4y^7z^3}{xy^{10}z^{-3}}$

Possible Answers:

None of the above

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

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

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

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

Correct answer:

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

Explanation:

To solve this problem you must recall that you need to subtract the exponents. Subtracting the smaller from the larger.

So:

$\frac{x^{4-1}z^3}{y^{10-7}z^{-3}}$

From here you need to get the negative exponent to be positive by bringing it to the top.

So:

$\frac{x^{4-1}z^{3+3}}{y^{10-7}}$

Then simplify:

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

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Example Question #41 : Simplifying Exponents

Multiply: $8^{\frac{2}{3}} \times 8^{\frac{3}{2}}$

Possible Answers:

$64\sqrt2$

$16\sqrt2$

$32\sqrt2$

Correct answer:

$64\sqrt2$

Explanation:

The bases of the exponents are common. This means we can add the fractions.

$8^{\frac{2}{3}} \times 8^{\frac{3}{2}} = 8^{\frac{2}{3}+ \frac{3}{2}}$

The least common denominator is six.

$\frac{2}{3}+ \frac{3}{2} = \frac{4}{6}+ \frac{9}{6} = \frac{13}{6}$

This becomes the power of the exponent.

$8^{\frac{13}{6}}$

Break up the fraction in terms so that each can be reduced.

$8^{\frac{13}{6}} = 8^{\frac{12}{6}} \cdot 8^{\frac{1}{6}} = 8^2 \cdot 8^{\frac{1}{6}}$

Since we do not know term $8^{\frac{1}{6}}$ , it can be rewritten in base two, and 8 = 2^3 .

Rewrite this term as a replacement of $8^{\frac{1}{6}}$ , and multiply the power of the exponent in base two with the power of the exponent in base eight.

$8^2\cdot 8^{\frac{1}{6}} = 8^2 \cdot (2^3)^{(\frac{1}{6})} = 8^2\cdot 2^{\frac{1}{2}}$

Simplify the terms. A value to the power of one-half is the square root of that number.

$8^2\cdot 2^{\frac{1}{2}} = 64\sqrt2$

The answer is: $64\sqrt2$

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Example Question #41 : Simplifying Exponents

Evaluate at: $\frac{x^{\frac{3}{4}}}{x^2}$

Possible Answers:

$x^{\frac{8}{3}}$

$\frac{1}{x^{\frac{5}{4}}}$

$\frac{1}{x^{\frac{4}{5}}}$

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

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

Correct answer:

$\frac{1}{x^{\frac{5}{4}}}$

Explanation:

The exponents can be subtracted since the bases are alike.

Subtract the top and bottom exponents.

$\frac{x^{\frac{3}{4}}}{x^2} = x^{\frac{3}{4}-2}$

$\frac{3}{4} - 2 = \frac{3}{4}-\frac{8}{4} = -\frac{5}{4}$

The exponent becomes $x^{-\frac{5}{4}}$ . Eliminate the negative by taking the reciprocal of the base with the positive power.

$x^{-\frac{5}{4}} = \frac{1}{x^{\frac{5}{4}}}$

The answer is: $\frac{1}{x^{\frac{5}{4}}}$

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Example Question #371 : Exponents

Simplify the following expression:

$\frac{15x^7}{5x^6}$

Possible Answers:

$3x^{13}$

10x

$15x^{13}$

Correct answer:

Explanation:

Simplify the following expression:

$\frac{15x^7}{5x^6}$

To simplify this expression, we need to divide the constant terms and subtract the exponents. In other words, we treat the whole numbers out in front just like a regular division problem and we subtract the exponents above the x's.

$\frac{15x^7}{5x^6}=\frac{15}{5}x^{7-6}=3x^1=3x$

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Example Question #42 : Multiplying And Dividing Exponents

Simplify: $\frac{9w^{-4}x^2}{18w^8x^{-9}}$

Possible Answers:

$\frac{x^{10}}{2w^{12}}$

$\frac{9x^{11}}{2w^{12}}$

$\frac{x^{11}}{2w^{12}}$

$-\frac{x^{11}}{2w^{12}}$

$\frac{x^{11}}{2w^{2}}$

Correct answer:

$\frac{x^{11}}{2w^{12}}$

Explanation:

To simplify this expression, I would start by rewriting the negative exponents as positive. To do so, you have to place the term in its opposite place (if it's in the denominator, it then goes in the numerator). When you do that, you get: $\frac{9x^2x^9}{18w^8w^4}$ . Now, simplify each term separately. 9 goes into both 9 and 18, so that can be simpified. To simplify the x and w's, remember that when multiplying and bases are the same, you add the exponents. Therefore, your answer is: $\frac{x^{11}}{2w^{12}}$ .

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Example Question #44 : Multiplying And Dividing Exponents

Simplify the following expression:

$\frac{8x^5}{4x^2}*7x^5$

Possible Answers:

14x^8

14x^6

9x^8

$9x^{15}$

Correct answer:

14x^8

Explanation:

Simplify the following expression:

$\frac{8x^5}{4x^2}*7x^5$

Let's begin by simplifying the fraction on the left.

When dividing exponents, we need to subtract them. treat the integers like you would any fraction:

$2x^{5-2}*7x^5=2x^{3}*7x^5$

Next, simplify the multiplications step by adding the exponents and treating the integers like regular multiplication:

$2x^{3}*7x^5=14x^8$

So our answer is:

14x^8

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Example Question #372 : Exponents

Simplify the following expression.

$\frac{25b^4}{5b^2}$

Possible Answers:

15b^3

5b^2

25b^4

5b^6

Correct answer:

5b^2

Explanation:

Simplify the following expression

$\frac{25b^4}{5b^2}$

To simplify this, we need to subtract our exponents and divide our whole numbers.

When we do this, we get the following.

$\frac{25b^4}{5b^2}=\frac{25}{5}\frac{b^4}{b^2}=\frac{5\cdot 5b^2\cdot b^2}{5b^2}=5b^2$

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Example Question #3505 : Algebra Ii

Simplify: $\frac{2x^3y^9z^5}{16xy^{13}z^2}$

Possible Answers:

$\frac{x^2z^3}{8y^2}$

$\frac{xz^3}{8y^4}$

$\frac{x^2z^3}{8y^4}$

$\frac{x^2}{8y^4}$

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

Correct answer:

$\frac{x^2z^3}{8y^4}$

Explanation:

To simplify this expression, look at each term separately. $\frac{2}{16}=\frac{1}{8}$ , $\frac{x^3}{x}=x^2$ , $\frac{y^9}{y^{13}}=\frac{1}{y^4}$ , and $\frac{z^5}{z^2}=z^3$ . Put those all together to get: $\frac{x^2z^3}{8y^4}$ .

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Example Question #373 : Exponents

What is the product of 3x^2 and 6x^4?

Possible Answers:

18x

18x^4

18x^6

18x^7

Correct answer:

18x^6

Explanation:

Remember, product means multiplication: (3x^2)(6x^4)=18x^6 . Multiply like terms and recall that when bases are the same, add exponents.

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Example Question #49 : Simplifying Exponents

What is the quotient of 24y^9 and $6y^{-4}$ ?

Possible Answers:

$4y^{-13}$

$4y^{3}$

$4y^{11}$

$4y^{13}$

$y^{13}$

Correct answer:

$4y^{13}$

Explanation:

Recall that quotient means division: $\frac{24y^9}{6y^{-4}}$ . To make the negative exponent positive, put it in the numerator: $\frac{24y^9y^4}{6}$ . Now, simplify. Remember that when multiplying and bases are the same, add exponents. Your answer is: $4y^{13}$ .

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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 #35 : Simplifying Exponents

Example Question #41 : Simplifying Exponents

Example Question #41 : Simplifying Exponents

Example Question #371 : Exponents

Example Question #42 : Multiplying And Dividing Exponents

Example Question #44 : Multiplying And Dividing Exponents

Example Question #372 : Exponents

Example Question #3505 : Algebra Ii

Example Question #373 : Exponents

Example Question #49 : Simplifying Exponents

All Algebra II Resources

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