Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[3]{\frac{3 x y^{2}}{81 x^{4} y^{2}}}$$

Short answer

\(\frac{1}{3x}\)

Step-by-step solution

- Simplify the fraction under the cube root

Simplify the fraction \(\frac{3 x y^{2}}{81 x^{4} y^{2}}\). Start by dividing the coefficients and then the variables. \(\frac{3}{81} = \frac{1}{27}\) and the variables \(\frac{x}{x^{4}}= \frac{1}{x^{3}}\). The \(y^{2}\) terms cancel each other: \(\frac{y^{2}}{y^{2}} = 1\). This gives us \(\frac{1}{27 x^{3}}\).

- Write the simplified expression under the cube root

Now the expression under the cube root is \(\frac{1}{27 x^{3}}\). So we need to simplify \(\root{3}{\frac{1}{27 x^{3}}}\).

- Apply the cube root to the numerator and the denominator

Apply the cube root separately to the numerator and the denominator: \(\root{3}{1}\) and \(\root{3}{27 x^{3}}\). \(\root{3}{1} = 1\) and \(\root{3}{27} = 3\). Also, since \(\root{3}{x^{3}} = x\), we have \(\root{3}{27 x^{3}} = 3x\).

- Write the final simplified expression

Combine the results of the cube root to get the simplified expression: \(\frac{1}{3x}\).

Key concepts

Simplifying Expressions

Simplifying expressions often means breaking down a complex expression into a more manageable and compact form. This makes it easier to work with and understand. For instance, knowing that the numerator and denominator in a fraction can sometimes be divided by the same number can be a powerful tool to simplify. Let's look at our example:

• The expression is \[ \sqrt[3]{\frac{3 x y^{2}}{81 x^{4} y^{2}}} \], and we aim to bring it to a simpler form.
• We start by simplifying the fraction under the cube root. Simplify the coefficients first, \[ \frac{3}{81} = \frac{1}{27} \].
• Next, simplify the variables. \[ \frac{x}{x^{4}} = \frac{1}{x^{3}} \], and \[ \frac{y^{2}}{y^{2}} = 1 \].
• Now, our fraction becomes \[ \frac{1}{27 x^{3}} \].

Simplified fractions and variables are steps that make a huge difference. Always start by reducing what you can to the simplest form.

Variables

Variables are symbols used to represent numbers or values in expressions and equations. In our example, we have variables like x and y. Here's a breakdown:

• When simplifying expressions with variables, treat the variables just like numbers—divide them when they appear in both the numerator and the denominator.
• In \[ \frac{3 x y^{2}}{81 x^{4} y^{2}} \], we divided the x and y variables as follows:
• For the x variable: \[ x^{1} / x^{4} = x^{1-4} = x^{-3} = \frac{1}{x^{3}} \]
• For the y variable: \[ y^{2} / y^{2} = 1 \]

Variables can be challenging, but simplifying them makes problems much easier. Always remember to use the properties of exponents to simplify correctly.

Fractions

Fractions represent parts of a whole and are expressed as \[ \frac{numerator}{denominator} \]. Simplifying fractions involves reducing them to their simplest form. In our example, we learned about this in a few ways:

• First, we divided the coefficients: \[ \frac{3}{81} \] which simplified to \[ \frac{1}{27} \].
• Next, we simplified the variables, yielding \[ \frac{x}{x^{4}} = \frac{1}{x^{3}} \].
• So our new simplified fraction was \[ \frac{1}{27 x^{3}} \].
• To further simplify, we applied the cube root to both the numerator and the denominator: \[ \sqrt[3]{1} = 1 \] and \[ \sqrt[3]{27 x^{3}} = 3x \].
• This gave our final simplified form: \[ \frac{1}{3x} \].

Reducing fractions and applying roots or other operations can simplify an expression significantly, making it easier to understand and work with.