Use rational exponents to simplify the expression. If rational exponents
appear after simplifying, write the answer in radical notation. Assume that
all variables represent positive numbers.

Ask by Mcdonald Clarke. in the United States

Dec 06,2024

Question

Use rational exponents to simplify the expression. If rational exponents
appear after simplifying, write the answer in radical notation. Assume that
all variables represent positive numbers.

Ask by Mcdonald Clarke. in the United States

Dec 06,2024

UpStudy AI · Accepted Answer

To simplify the expression $$ \frac{12}{x} \sqrt[13]{x} $$ using rational exponents, we first rewrite the square root in terms of exponents.

The expression can be rewritten as:

$$
\frac{12}{x} \cdot x^{1/13}
$$

Now, we can combine the terms involving $$ x $$:

$$
\frac{12 \cdot x^{1/13}}{x}
$$

Using the property of exponents $$ \frac{a^m}{a^n} = a^{m-n} $$, we can simplify $$ x^{1/13} $$ divided by $$ x^1 $$:

$$
\frac{12 \cdot x^{1/13}}{x^1} = 12 \cdot x^{1/13 - 1} = 12 \cdot x^{1/13 - 13/13} = 12 \cdot x^{-12/13}
$$

Now, we can express $$ x^{-12/13} $$ in radical notation:

$$
12 \cdot \frac{1}{x^{12/13}} = 12 \cdot \frac{1}{\sqrt[13]{x^{12}}}
$$

Thus, the final simplified expression in radical notation is:

$$
\frac{12}{\sqrt[13]{x^{12}}}
$$
The simplified expression is $$ \frac{12}{\sqrt[13]{x^{12}}} $$.