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Precalculus : Rational Exponents

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Example Questions

Example Question #1141 : Pre Calculus

Simplify the function:

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

Possible Answers:

y =(2x+7)

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

y =(2x+7)^3

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

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 #1151 : Pre Calculus

Simplify the function:

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

Possible Answers:

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

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

y = (-7x+12)^7

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

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 #1152 : Pre Calculus

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:

$\frac{8}{25}$

$\frac{16}{5}$

$\frac{8}{5}$

$\frac{16}{25}$

$\frac{2}{5}$

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 #1153 : Pre Calculus

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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Example Question #1154 : Pre Calculus

Evaluate the following expression and solve for .

$343=7^{15t-27}$

Possible Answers:

t=3

t=-2

t=2

t=7

$t=\sqrt{27}$

Correct answer:

t=2

Explanation:

To solve this problem, recall that you can set exponents equal to eachother if they have the same base.

See below:

343=7^3

So, we have

$7^3=7^{15t-27}$

Because both sides of this equation have a base of seven, we can set the exponents equal to eachother and solve for t.

3=15t-27

30=15t

t=2

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Example Question #1155 : Pre Calculus

Solve for .

$2^{x/3}\cdot 3^{x/4}\cdot e^{-x}=3$

Possible Answers:

$x=\frac{36}{\ln432}-1\approx4.932$

$x=\frac{3}{\ln432e^{-12}}\approx-0.506$

$x=\frac{36}{\ln432e^{-12}}\approx-6.069$

$x=\frac{\ln3}{\ln(2^{1/3}\cdot3^{1/4}\cdot e^{-1})}\approx-2.223$

Correct answer:

$x=\frac{\ln3}{\ln(2^{1/3}\cdot3^{1/4}\cdot e^{-1})}\approx-2.223$

Explanation:

We begin by taking the natural log of the equation:

$\ln(2^{x/3}\cdot 3^{x/4}\cdot e^{-x})=\ln3$

Simplifying the left side of the equation using the rules of logarithms gives:

$\frac{x}{3}\ln2+\frac{x}{4}\ln3-x=\ln3$

We group the x terms to get:

$x\left(\frac{1}{3}\ln2+\frac{1}{4}\ln3-1\right)=\ln3$

We reincorporate the exponents into the logarithms and use the identity property of the natural log to obtain:

$x(\ln2^{1/3}+\ln3^{1/4}+\ln e^{-1})=\ln3$

We combine the logarithms using the multiplication/sum rule to get:

$x\ln(2^{1/3} 3^{1/4}e^{-1})=\ln3$

We then solve for x:

$x=\frac{\ln3}{\ln(2^{1/3}3^{1/4}e^{-1})}\approx-2.223$

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Example Question #63 : Exponential And Logarithmic Functions

Solve for .

$x^{5/2}-12x^{3/2}+20x^{1/2}=0$

Possible Answers:

x=0,4,5

x=2,10

x=0,2,10

x=0

Correct answer:

x=0,2,10

Explanation:

We begin by factoring out the term $x^{1/2}$ to get:

$x^{1/2}(x^2-12x+20)=0$

This equation gives our first solution:

$x^{1/2}=0\Rightarrow x=0$

Then we check for more solutions:

$\\x^2-12x+20=0 \\ \Rightarrow (x-10)(x-2)=0\\ \Rightarrow x=2,10$

Therefore our solution is

x=0,2,10

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Example Question #1156 : Pre Calculus

Evaluate $\frac{3xz^{\frac{1}{3}}}{(6x)^{\frac{1}{2}}y^{2}z^{3}}$ when $x=2,y=3^{\frac{1}{4}}, z=8$

Possible Answers:

$\frac{1}{256}$

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

$\frac{3\sqrt{8}}{2^{8}\sqrt{6}}$

$\frac{\sqrt{3}}{ 2^{8}\sqrt{2}}$

Correct answer:

$\frac{1}{256}$

Explanation:

Remember the denominator of a rational exponent is equivalent to the index of a root.

This should simplify quite nicely.

$\frac{3xz^{\frac{1}{3}}}{(6x)^{\frac{1}{2}}y^{2}z^{3}}=\frac{3x^{1/2}}{\sqrt{2\cdot3}y^2z^{8/3}}=\frac{\sqrt{3x}}{\sqrt{2}y^2\sqrt[3]{z^{8}}}$

When $x=2,y=3^{\frac{1}{4}}, z=8$ it gives us,

$\frac{\sqrt{3\cdot2}}{\sqrt{2}(3^{\frac{1}{4}})^2\sqrt[3]{8^{8}}}=\frac{\sqrt{3}}{(3^{\frac{2}{4}})2^{8}}=\frac{1}{2^8}=\frac{1}{256}$

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Example Question #65 : Exponential And Logarithmic Functions

What is the value of $125^\frac{1}{3}$ ?

Possible Answers:

122

41.7

Correct answer:

Explanation:

What does an exponent of one-third mean? Consider our expression and raise it to the third power.

$(125^\frac{1}{3})^3 = x^3$

Simplifying, we get:

125 = x^3

Thus, we are looking for a number that when cubed, we get 125 . Thus, we are discussing the cube root of 125 , or .

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Precalculus : Rational Exponents

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1141 : Pre Calculus

Example Question #1151 : Pre Calculus

Example Question #1152 : Pre Calculus

Example Question #1153 : Pre Calculus

Example Question #1154 : Pre Calculus

Example Question #1155 : Pre Calculus

Example Question #63 : Exponential And Logarithmic Functions

Example Question #1156 : Pre Calculus

Example Question #65 : Exponential And Logarithmic Functions

All Precalculus Resources

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