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Precalculus : Simplify Expressions With Rational Exponents

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Example Question #1 : Simplify Expressions With Rational Exponents

Simplify

$\frac{9^{\frac{2}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}$

Possible Answers:

$\frac{9}{4}$

$\frac{9}{2}$

$\frac{3}{4}$

$\frac{27}{2}$

Correct answer:

$\frac{3}{4}$

Explanation:

$\frac{9^{\frac{2}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}=\frac{(3^{2})^{\frac{2}{3}}\cdot3^{\frac{5}{3}}}{6^{2}}=\frac{3^{\frac{4}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}$

$=\frac{3^\frac{9}{3}}{6^2}=\frac{3^3}{6^2}=\frac{27}{36}=\frac{3}{4}$ .

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Example Question #1 : Simplify Expressions With Rational Exponents

Simplify the expression.

$\frac{x^{3/2}y^{7/3}}{x^3 y^{2/9}}$

Possible Answers:

$\frac{1}{x^{9/2}y^{14/27}}$

$\frac{y^{19/9}}{x^{3/2}}$

$x^{3/2}y^{19/9}$

$\frac{x^{3/2}}{y^{19/9}}$

$x^{9/2}y^{23/9}$

Correct answer:

$\frac{y^{19/9}}{x^{3/2}}$

Explanation:

Using the properties of exponents, we can either choose to subtract the exponents of the corresponding bases or rewrite the expression using negative exponents as such:

$x^{3/2}x^{-3}y^{7/3}y^{-2/9}$

Here, we combine the terms with corresponding bases by adding the exponents together to get

$x^{-3/2}y^{19/9}$

Placing the x term (since it has a negative exponent) in the denominator will result in the correct answer. It can be shown that simply subtracting the exponents of corresponding bases will result in the same answer.

$\frac{y^{19/9}}{x^{3/2}}$

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Example Question #3 : Simplify Expressions With Rational Exponents

Simplify the expression $4^{2.5}$ .

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We proceed as follows

$4^{2.5}$

Write 2.5 as a fraction

$4^{5/2}$

The denominator of the fraction is a , so it becomes a square root.

$(^2\sqrt{4})^5$

Take the square root.

2^5

Raise to the 5th power.

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Example Question #4 : Simplify Expressions With Rational Exponents

What is the value of $32^{\frac{3}{5}}$ ?

Possible Answers:

19.2

Correct answer:

Explanation:

Recall that when considering rational exponents, the denominator of the fraction tells us the "root" of the expression.

Thus in this case we are taking the fifth root of .

The fifth root of is , because $2\cdot2\cdot2\cdot2\cdot2 = 32$ .

Thus, we have reduced our expression to $2^3 = 2\cdot2\cdot2 = 8$ .

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Example Question #5 : Simplify Expressions With Rational Exponents

Simplify the expression: $\frac{2x^3y^4}{6x^4y}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

Simplify the constants:

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

Subtract the "x" exponents:

$\frac{y^4}{3xy}$

$x^3-x^4=x^{3-4}=x^{-1}=\frac{1}{x}$ This is how the x moves to the denominator.

Finally subtract the "y" exponents:

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

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Example Question #6 : Simplify Expressions With Rational Exponents

Solve:

$3x^{\frac{3}{4}} = x^{\frac{1}{2}}$

Possible Answers:

$x = \frac{3}{4}$

x = 3

x = 9

$x = \frac{1}{81}$

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

Correct answer:

$x = \frac{1}{81}$

Explanation:

To remove the fractional exponents, raise both sides to the second power and simplify:

$(3x^{\frac{3}{4}})^{2} = (x^{\frac{1}{2}})^{2}$

$9x^{\frac{3}{2}} = x$

$\frac{9x^{\frac{3}{2}}}{x} = \frac{x}{x}$

$9x^{\frac{1}{2}} = 1$

Now solve for :

$\frac{9x^{\frac{3}{2}}}{9} = \frac{1}{9}$

$x^{\frac{1}{2}} = \frac{1}{9}$

$(x^{\frac{1}{2}})^{2} = (\frac{1}{9})^{2}$

$x = \frac{1}{81}$

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Example Question #7 : Simplify Expressions With Rational Exponents

Solve:

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

Possible Answers:

x = 6

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

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

x = 1

x = 8

Correct answer:

x = 8

Explanation:

To remove the rational exponent, cube both sides of the equation:

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

Now simplify both sides of the equation:

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

$x^{1} = 2^{3}$

x = 8

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Example Question #8 : Simplify Expressions With Rational Exponents

Simplify and rewrite with positive exponents:

$\frac{2a^3b^{28}c^2d^2e^4f}{3a^2b^{27}cd^3e^5f^2}$

Possible Answers:

$\frac{2abc}{3def}$

$\frac{2abcd^{-1}e^{-1}f^{-1}}{3}$

$\frac{2def}{3abc}$

$\frac{2a^6bc^3}{3def}$

$\frac{2def}{3a^6bc^3}$

Correct answer:

$\frac{2abc}{3def}$

Explanation:

$\frac{2a^3b^{28}c^2d^2e^4f}{3a^2b^{27}cd^3e^5f^2}$

When dividing two exponents with the same base we subtract the exponents:

$\frac{2abcd^{-1}e^{-1}f^{-1}}{3}$

Negative exponents are dealt with based on the rule

$a^{-m}=\frac{1}{a^m}$ :

$\mathbf{\frac{2abc}{3def}}$

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Example Question #1 : Simplify Expressions With Rational Exponents

Simplify the function:

$y = ((x+2)^{1/2})^2$

Possible Answers:

$y = (x+2)^{3}$

$y = (x+2)^{1/4}$

$y = (x+2)^{3/2}$

y = (x+2)

Correct answer:

y = (x+2)

Explanation:

When an exponent is raised to the power of another exponent, just multiply the exponents together.

$y = (x^a)^b = x^{ab}$

$y = ((x+2)^{1/2})^2 = (x+2)^1$

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Example Question #1 : Simplify Expressions With Rational Exponents

Simplify:

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

Possible Answers:

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

4xy^4

4xy^6

None of the other answers.

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

Correct answer:

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

Explanation:

Subtract the "x" exponents and the "y" exponents vertically. Then add the exponents horizontally if they have the same base (subtract the "x" and subtract the "y" ones). Finally move the negative exponent to the denominator.

$\frac{8xy^2}{2x^4y}*x^2y^3\rightarrow 4x^{-3}y*x^2y^3\rightarrow 4x^{-1}y^4=\frac{4y^4}{x}$

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Precalculus : Simplify Expressions With Rational Exponents

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