Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Simplify. Assume all variables are positive. $\newline$ $z^{\frac{2}{3}} \cdot z^{\frac{5}{3}}$ $\newline$ Write your answer in the form $A$ or $\frac{A}{B}$ , where $A$ and $B$ are constants or variable expressions that have no variables in common. All exponents in your answer should be positive. $\newline$ ______

View step-by-step help

View step-by-step help

Simplify expressions involving rational exponents

Full solution

Q. Simplify. Assume all variables are positive.

\newline

z^{\frac{2}{3}} \cdot z^{\frac{5}{3}}

\newline

Write your answer in the form

A

or

\frac{A}{B}

, where

A

and

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

______

Identify Equation and Apply Property: Identify the equation and apply the property of exponents for multiplication, which states that when multiplying like bases, you add the exponents: $a^m \times a^n = a^{m+n}$ . So, $z^{\frac{2}{3}} \times z^{\frac{5}{3}} = z^{\frac{2}{3} + \frac{5}{3}}$ .
Add Exponents: Add the exponents $\frac{2}{3}$ and $\frac{5}{3}$ . $\newline$ $\frac{2}{3} + \frac{5}{3} = \frac{(2+5)}{3} = \frac{7}{3}$ . $\newline$ So, $z^{\frac{2}{3}} * z^{\frac{5}{3}} = z^{\frac{7}{3}}$ .
Check Final Exponent: Check that the final exponent is positive, which it is $7/3$ , and that there are no variables in common in the numerator and denominator of the exponent, which is also true. $\newline$ Therefore, the simplified form of the expression is $z^{7/3}$ .

More problems from Simplify expressions involving rational exponents

Question

f(x)=\frac{6 x+5}{2-\sqrt{14+5 x}}

\newline

We want to find

\lim _{x \rightarrow-2} f(x)

\newline

What happens when we use direct substitution?

\newline

1

\newline

(A) The limit exists, and we found it!

\newline

(B) The limit doesn't exist (probably an asymptote).

\newline

(C) The result is indeterminate.

Posted 9 months ago

Question

x^{4}+y^{2}=17

\newline

What is the value of

\frac{d^{2} y}{d x^{2}}

(-2,1)

\newline

Give an exact number.

Posted 8 months ago

Question

What is the area of the region bound by the graphs of

f(x)=\sqrt{x-2}, g(x)=14-x

x=2

\newline

1

\newline

\frac{19}{6}

\newline

\frac{99}{2}

\newline

\frac{151}{2}

\newline

\frac{45}{2}

Posted 7 months ago

Question

g(x)=x^{\frac{4}{3}}

\newline

g^{\prime}(27)=

Posted 8 months ago

Question

\ln \left(\frac{1}{\sqrt{e}}\right)

\newline

Posted 9 months ago

Question

\ln \left(\frac{1}{e}\right)

\newline

Posted 9 months ago

Question

y=(\sin^{-1}(5x^{2}))^{3}

Posted 8 months ago

Question

\left(10x^{3}-5x^{2}-6x\right)\div\left(2x^{2}\right)

Posted 8 months ago

Question

y=\cos^{4}x\sin^{4}x

Posted 8 months ago

Question

\frac{3}{5}-\frac{3}{10}

Posted 8 months ago

Related topics

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