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Precalculus : Exponential and Logarithmic Functions

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Example Questions

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 #2 : Simplify Expressions With Rational Exponents

Solve:

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

Possible Answers:

x = 6

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

x = 1

x = 8

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

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 #2 : Rational Exponents

Simplify the function:

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

Possible Answers:

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

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

y = (x+2)

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

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:

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

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

None of the other answers.

4xy^4

4xy^6

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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Example Question #51 : Exponential And Logarithmic Functions

Simplify the function:

$y = [((2x+7)^{1/4})^3]^4$

Possible Answers:

y =(2x+7)

$y =(2x+7)^{29/4}$

$y =(2x+7)^{7/4}$

y =(2x+7)^3

Correct answer:

y =(2x+7)^3

Explanation:

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

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

$y = [((2x+7)^{1/4})^3]^4 = ((2x+7)^{1/4})^{12} = (2x+7)^3$

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Example Question #52 : Exponential And Logarithmic Functions

Simplify the function:

$y = [((-7x+12)^{1/3})^3]^7$

Possible Answers:

$y = (-7x+12)^{\frac{10}{3}}$

$y = (-7x+12)^{\frac{16}{3}}$

$y = (-7x+12)^{\frac{8}{3}}$

y = (-7x+12)^7

Correct answer:

y = (-7x+12)^7

Explanation:

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

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

$y = [((-7x+12)^{1/3})^3]^7 = ((-7x+12)^{1/3})^{21}$ = (-7x+12)^7

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Example Question #53 : Exponential And Logarithmic Functions

Simplify the expression:

$\frac{2^\frac{5}{3}\cdot 4^\frac{2}{3}\cdot 8^\frac{1}{3}}{5^\frac{1}{3}\cdot 25^\frac{1}{3}}$ .

Possible Answers:

$\frac{16}{5}$

$\frac{2}{5}$

$\frac{8}{25}$

$\frac{16}{25}$

$\frac{8}{5}$

Correct answer:

$\frac{16}{5}$

Explanation:

First, you can begin to simplfy the numerator by converting all 3 expressions into base 2.

$2^\frac{5}{3}\cdot (2^2)^\frac{4}{3}\cdot (2^3)^\frac{1}{3}$ , which simplifies to

$2^3\cdot 2 =16$

For the denominator, the same method applies. Convert the 25 into base 5, and when simplified becomes simply 5.

$5^{\frac{1}{3}}\cdot 5^{2\cdot \frac{1}{3}}$

$5^{\frac{1}{3}+\frac{2}{3}}=5^1=5$

The final simplified answer becomes:

$\frac{\text{Numerator}}{\text{Denominator}} = \frac{16}{5}$

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Example Question #14 : Simplify Expressions With Rational Exponents

Evaluate the following expression using knowledge of the properties of exponents:

$8^\frac{2}{3}+(2^2)^2-(3)^3\left(\frac{1}{3}\right)^3+\frac{5^{19}}{5^{17}}-647^0$

Possible Answers:

674

Correct answer:

Explanation:

Let's work through this equation involving exponents one term at a time. The first term we see is $8^\frac{2}{3}$ , for which we can apply the following property:

$a^\frac{c}{b}=\sqrt[b]{a^c}=(\sqrt[b]{a})^c$

So if we plug our values into the formula for the property, we get:

$8^\frac{2}{3}=\sqrt[3]{8^2}=(\sqrt[3]{8})^2=(2)^2=4$

Because 2^3=8 . Our next term is (2^2)^2 , for which we'll need the property:

$(a^b)^c=a^{bc}$

Using the values for our term, we have:

$(2^2)^2=2^4=2\cdot 2\cdot 2\cdot 2=16$

The third term of the equation is $(3)^3\left(\frac{1}{3}\right)^3$ , for which the quickest way to evaluate would be using the following property:

(a)^c(b)^c=(ab)^c

Using the values from our term, this gives us:

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

The next property we will need to consider for our fourth term is given below:

$\frac{a^b}{a^c}=a^{b-c}$

If we plug in the corresponding values from our term, we get:

$\frac{5^{19}}{5^{17}}=5^{19-17}=5^2=25$

Finally, our last term requires knowledge of the following simple property: Any number raised to the power of zero is 1. With this in mind, our last term becomes:

647^0=1

Rewriting the equation with all of the values we've just evaluated, we obtain our final answer:

4+16-1+25-1=43

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Precalculus : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #5 : Simplify Expressions With Rational Exponents

Example Question #6 : Simplify Expressions With Rational Exponents

Example Question #2 : Simplify Expressions With Rational Exponents

Example Question #8 : Simplify Expressions With Rational Exponents

Example Question #2 : Rational Exponents

Example Question #1 : Simplify Expressions With Rational Exponents

Example Question #51 : Exponential And Logarithmic Functions

Example Question #52 : Exponential And Logarithmic Functions

Example Question #53 : Exponential And Logarithmic Functions

Example Question #14 : Simplify Expressions With Rational Exponents

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