Question

simplify

\\\frac{1}{\sqrt3{2x}}\\

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Identify the expression

The given expression is:

$$ \frac{1}{\sqrt[3]{2x}} $$

To simplify this expression, we need to rationalize the denominator. This means we want to eliminate the radical (the cube root) from the bottom of the fraction.

Step 2: Determine the rationalizing factor

For a cube root in the denominator, we need the term inside the radical to be a perfect cube so that the cube root simplifies to a rational expression.

The current term inside the cube root is \( 2x \), which is \( 2^1 \cdot x^1 \). To make it a perfect cube, we need each factor to have an exponent of 3.

• We need two more factors of 2: \( 2^2 = 4 \)
• We need two more factors of \( x \): \( x^2 \)

Thus, we multiply both the numerator and the denominator by \( \sqrt[3]{4x^2} \):

$$ \frac{1}{\sqrt[3]{2x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} $$

Step 3: Multiply and simplify

Multiply the numerators:

$$ 1 \cdot \sqrt[3]{4x^2} = \sqrt[3]{4x^2} $$

Multiply the denominators:

$$ \sqrt[3]{2x} \cdot \sqrt[3]{4x^2} = \sqrt[3]{8x^3} $$

Since \( 8x^3 \) is a perfect cube, we can simplify the denominator:

$$ \sqrt[3]{8x^3} = 2x $$

Putting it all together:

$$ \frac{\sqrt[3]{4x^2}}{2x} $$

Answer:

$$ \frac{\sqrt[3]{4x^2}}{2x} $$