Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

root(3)(x^(2))*x^(-(1)/(5))
Which of the following is equivalent to the given expression for all real values of
x ?
Choose 1 answer:
(A)
root(15)(x^(7))
(B)
root(7)(x^(15))
(C)
(1)/(sqrt(x^(15)))
(D)
(1)/(root(15)(x^(2)))

$\sqrt[3]{x^{2}} \cdot x^{-\frac{1}{5}}$ $\newline$ Which of the following is equivalent to the given expression for all real values of $x$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\sqrt[15]{x^{7}}$ $\newline$ (B) $\sqrt[7]{x^{15}}$ $\newline$ (C) $\frac{1}{\sqrt{x^{15}}}$ $\newline$ (D) $\frac{1}{\sqrt[15]{x^{2}}}$

View step-by-step help

View step-by-step help

Compare linear and exponential growth

Full solution

Q.

\sqrt[3]{x^{2}} \cdot x^{-\frac{1}{5}}

\newline

Which of the following is equivalent to the given expression for all real values of

x

?

\newline

Choose

1

answer:

\newline

(A)

\sqrt[15]{x^{7}}

\newline

(B)

\sqrt[7]{x^{15}}

\newline

(C)

\frac{1}{\sqrt{x^{15}}}

\newline

(D)

\frac{1}{\sqrt[15]{x^{2}}}

Given Expression: We are given the expression $\sqrt[3]{x^2} \cdot x^{-\frac{1}{5}}$ . To simplify this expression, we need to combine the exponents by using the properties of exponents.
Convert Cube Root to Exponent: First, we convert the cube root into an exponent. The cube root of $x^2$ can be written as $x^{\frac{2}{3}}$ . So the expression becomes $x^{\frac{2}{3}} \cdot x^{-\frac{1}{5}}$ .
Add Exponents of Same Base: Next, we add the exponents of the same base $x$ when multiplying. The sum of the exponents is $\frac{2}{3} + (-\frac{1}{5})$ . To add these fractions, we need a common denominator, which is $15$ . So we convert the fractions: $\frac{2}{3} = \frac{10}{15}$ and $-\frac{1}{5} = -\frac{3}{15}$ .
Convert Exponent Back to Radical Form: Now we add the converted exponents: $\frac{10}{15} + (-\frac{3}{15}) = \frac{10 - 3}{15} = \frac{7}{15}$ . Therefore, the simplified expression is $x^{\frac{7}{15}}$ .
Match with Answer Choices: Finally, we convert the exponent back to a radical form. The expression $x^{\frac{7}{15}}$ is equivalent to the $15$ th root of $x^7$ , which is written as $\sqrt[15]{x^7}$ .
Match with Answer Choices: Finally, we convert the exponent back to a radical form. The expression $x^{\frac{7}{15}}$ is equivalent to the $15$ th root of $x^7$ , which is written as $\sqrt[15]{x^7}$ .Comparing the simplified expression with the answer choices, we find that it matches with choice (A) $\sqrt[15]{x^7}$ .

More problems from Compare linear and exponential growth

Question

Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits

x

days to pick up the motorcycle.

\newline

Write an equation for the function. If it is linear, write it in the form

g(x) = mx + b

. If it is exponential, write it in the form

g(x) = a(b)^x

\newline

g(x) = \underline{\hspace{3em}}

Posted 5 months ago

Question

f(t)=-2t-7

change over the interval from

t=-7

t=-6

\newline

\newline

\left[\text{f(t) decreases by } 2\right]

\newline

\left[\text{f(t) increases by } 2\right]

\newline

\left[\text{f(t) increases by } 200\%\right]

\newline

\left[\text{f(t) decreases by a factor of } 2\right]

Posted 5 months ago

Question

g(t)=9^t

change over the interval from

t=1

t=3

\newline

\newline

\left[\text{g(t) decreases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by } 18\%\right]

\newline

\left[\text{g(t) decreases by } 9\%\right]

Posted 5 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 6 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 6 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 6 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

Posted 7 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

u'(5)= -2

v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 7 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 7 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

n'(7)= 8

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 7 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant