Problem 70 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{\frac{4}{5 x}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\sqrt[3]{4} \times \sqrt[3]{x^2}}{5}\).

Step 1: Express the Cube Root as a Fractional Exponent

Firstly, remember that the cube root of a number is equivalent to raising that number to the power of \(\frac{1}{3}\). Therefore, the cube root of \(\frac{4}{5x}\) is equivalent to \(\left( \frac{4}{5x} \right) ^{\frac{1}{3}}\).

Step 2: Distribute the Exponent to the Numerator and Denominator

Next, distribute the \(\frac{1}{3}\) exponent across the numerator and denominator inside the brackets, that is, \(\left( \frac{4}{5x} \right) ^{\frac{1}{3}}\) becomes \(\frac{4^{\frac{1}{3}}}{(5x)^{\frac{1}{3}}}\).

Step 3: Rationalize the Denominator

In order to get rid of the cube root from the denominator (rationalize the denominator), you will multiply the expression by \(\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}\). Here, \(\frac{2}{3}\) is chosen because when added to \(\frac{1}{3}\), it yields 1, effectively removing the cube root from the denominator. Thus, the expression becomes \(\frac{4^{\frac{1}{3}} \times x^{\frac{2}{3}}}{5}\).

Step 4: Simplify Final Expression

Finally, rewrite the final expression with the fractional exponents simplified as a cube root to yield the final simplified expression: \(\frac{\sqrt[3]{4} \times \sqrt[3]{x^2}}{5}\).

Key Concepts

Fractional Exponents

Fractional exponents may seem complex at first, but they provide a useful way to represent roots in equations. Traditional exponents tell us how many times to multiply a number by itself. When the exponent is a fraction, it indicates a root. For example, the expression \( a^{1/3} \) represents the cube root of \( a \). In general:

• The denominator of the fraction (bottom number) tells us the root we are dealing with. For instance, in \( a^{1/3} \), \'3\' indicates a cube root.
• The numerator (top number) describes the power to which the base number is raised before the root is applied. So \( a^{2/3} \) means you square \( a \) and then take the cube root.

Understanding fractional exponents will help simplify expressions involving roots.This concept allows us to express and manipulate roots easily within algebraic operations.

Cube Roots

Cube roots are essential components in algebra, especially when dealing with volume or simplifying expressions. A cube root asks the question: 'What number, when multiplied by itself three times, gives the original number?' Cubic expressions are especially relevant in geometry.
To indicate a cube root mathematically, we use the notation \( \sqrt[3]{a} \), which is the same as \( a^{1/3} \). Here's how they are connected:

• \( \sqrt[3]{8} = 2 \) since \( 2 \times 2 \times 2 = 8 \).
• Consequently, \( 8^{1/3} = 2 \).

Cube roots are also important in physics and engineering, for calculating things like cube volumes or transformations in three dimensions. Rationalizing expressions containing cube roots simplifies calculations and clarifies results.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics that involves reducing an expression to its most basic form without changing its value. This makes the expression easier to work with, especially in complex equations. One common method to simplify expressions is by using fractional exponents to manage roots efficiently, as seen in:

• Converting \( \sqrt[3]{\frac{4}{5x}} \) into its fractional exponent form: \( \left( \frac{4}{5x} \right) ^{1/3} \).
• Applying the exponent individually to both the numerator and denominator: \( \frac{4^{1/3}}{(5x)^{1/3}} \).

Lastly, rationalizing the denominator is a process where you eliminate any roots by multiplying by a conjugate or a convenient form like \( x^{2/3} \) when dealing with cube roots, rephrasing the expression in a more manageable form.