$\frac{3}{\sqrt[3]{9 x}}$
Answer (1)
Rewrite the expression using exponents: 3 9 x 3 = ( 9 x ) 3 1 3 .
Distribute the exponent: ( 3 2 x ) 3 1 3 = 3 3 2 x 3 1 3 .
Simplify the expression: 3 3 2 x 3 1 3 1 = 3 3 1 x − 3 1 .
Rewrite the expression using radicals: 3 3 1 x − 3 1 = 3 x 3 .
The simplified expression is 3 x 3 .
Explanation
Understanding the Problem We are given the expression 3 9 x 3 and our goal is to simplify it.
Rewriting with Exponents First, let's rewrite the expression using exponents. Recall that n a = a n 1 . So, we have 3 9 x 3 = ( 9 x ) 3 1 3 We can also write 9 as 3 2 , so the expression becomes \frac{3}{(3^2x)^{\frac{1}{3}}}$ 3. Distributing the Exponent Now, let's distribute the exponent in the denominator: \frac{3}{(3^2x)^{\frac{1}{3}}} = \frac{3}{3^{\frac{2}{3}}x^{\frac{1}{3}}} We can rewrite $3$ in the numerator as $3^1$, so we have \frac{3^1}{3^{\frac{2}{3}}x^{\frac{1}{3}}}$
Simplifying the Expression Now, we can simplify the expression by dividing the powers of 3. Recall that a n a m = a m − n . So, we have 3 3 2 x 3 1 3 1 = 3 1 − 3 2 x − 3 1 = 3 3 1 x − 3 1 We can rewrite this as 3^{\frac{1}{3}} x^{-\frac{1}{3}} = \frac{3^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ 5. Rewriting with Radicals Finally, we can rewrite the expression using radicals again: \frac{3^{\frac{1}{3}}}{x^{\frac{1}{3}}} = \sqrt[3]{\frac{3}{x}} So, the simplified expression is 3 x 3 .
Final Answer Therefore, the simplified form of the given expression is 3 x 3 .
Examples
Imagine you are designing a container, and its volume is given by V = 3 9 x 3 . If you want to understand how the volume changes as x changes, simplifying the expression to V = 3 x 3 makes it easier to analyze. For example, you can quickly see that as x increases, the volume V decreases, and vice versa. This kind of simplification is useful in various engineering and design applications.
