Rationalize the Denominator cube root of 2/4

Solution:

Step 1: Simplify the Fraction

Reduce the common factors in the numerator and denominator.

Step 1.1: Factor Out Common Factors

Extract the factor of 2 from the numerator. $\sqrt[3]{\frac{2 \cdot 1}{4}}$

Step 1.2: Eliminate Common Factors

Step 1.2.1: Factor Out from Denominator

Factor 2 out of 4. $\sqrt[3]{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.2.2: Cancel Out Common Factors

Remove the common factor of 2. $\sqrt[3]{\frac{\cancel{2} \cdot 1}{\cancel{2} \cdot 2}}$

Step 1.2.3: Rewrite the Simplified Expression

The expression becomes $\sqrt[3]{\frac{1}{2}}$.

Step 2: Express as Individual Cube Roots

Split the cube root of the fraction into separate cube roots. $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$

Step 3: Simplify the Numerator

Recognize that the cube root of 1 is 1. $\frac{1}{\sqrt[3]{2}}$

Step 4: Rationalize the Denominator

Multiply by the conjugate cube root over itself. $\frac{1}{\sqrt[3]{2}} \cdot \frac{\left(\sqrt[3]{2}\right)^{2}}{\left(\sqrt[3]{2}\right)^{2}}$

Step 5: Combine and Simplify the Denominator

Step 5.1: Multiply Numerator and Denominator

Multiply the terms. $\frac{\left(\sqrt[3]{2}\right)^{2}}{\sqrt[3]{2} \cdot \left(\sqrt[3]{2}\right)^{2}}$

Step 5.2: Apply Exponent to the Cube Root

Raise the cube root to the power of 1. $\frac{\left(\sqrt[3]{2}\right)^{2}}{\left(\sqrt[3]{2}\right)^{1} \cdot \left(\sqrt[3]{2}\right)^{2}}$

Step 5.3: Combine Exponents

Use the exponent rule to combine terms. $\frac{\left(\sqrt[3]{2}\right)^{2}}{\left(\sqrt[3]{2}\right)^{1 + 2}}$

Step 5.4: Simplify the Exponent

Add the exponents. $\frac{\left(\sqrt[3]{2}\right)^{2}}{\left(\sqrt[3]{2}\right)^{3}}$

Step 5.5: Convert to Exponential Form

Step 5.5.1: Rewrite Cube Root as Exponent

Express the cube root as an exponent. $\frac{\left(\sqrt[3]{2}\right)^{2}}{\left(2^{\frac{1}{3}}\right)^{3}}$

Step 5.5.2: Apply Exponent Rule

Multiply the exponents. $\frac{\left(\sqrt[3]{2}\right)^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 5.5.3: Simplify the Exponent

Combine the terms. $\frac{\left(\sqrt[3]{2}\right)^{2}}{2^{\frac{3}{3}}}$

Step 5.5.4: Cancel the Common Factor

Step 5.5.4.1: Cancel the 3s

Remove the common factor of 3. $\frac{\left(\sqrt[3]{2}\right)^{2}}{2^{\frac{\cancel{3}}{\cancel{3}}}}$

Step 5.5.4.2: Rewrite the Expression

The expression simplifies to $\frac{\left(\sqrt[3]{2}\right)^{2}}{2}$.

Step 6: Simplify the Numerator

Step 6.1: Rewrite the Squared Cube Root

Express the squared cube root as a cube root of a square. $\frac{\sqrt[3]{2^2}}{2}$

Step 6.2: Calculate the Square

Square the number 2. $\frac{\sqrt[3]{4}}{2}$

Step 7: Present the Result in Different Forms

Exact Form: $\frac{\sqrt[3]{4}}{2}$ Decimal Form: $0.79370052 \ldots$

Knowledge Notes:

1. Rationalizing the Denominator: This process involves removing the radical (in this case, a cube root) from the denominator of a fraction. This is typically done by multiplying the fraction by a form of 1 that contains the radical in both the numerator and the denominator, thus eliminating the radical in the denominator after simplification.

2. Cube Root: The cube root of a number $a$, denoted as $\sqrt[3]{a}$, is a value that, when multiplied by itself three times, gives the number $a$. The cube root of 1 is always 1.

3. Exponent Rules: When multiplying like bases, you add the exponents: $a^m \cdot a^n = a^{m+n}$. When raising a power to a power, you multiply the exponents: $(a^m)^n = a^{mn}$.

4. Simplifying Fractions: To simplify a fraction, you divide the numerator and the denominator by their greatest common factor.

5. Conjugate Cube Roots: To rationalize a cube root in the denominator, you multiply by the square of the cube root over itself, which is the conjugate in terms of cube roots. This is because $(\sqrt[3]{a})^3 = a$, which eliminates the cube root when in the denominator.

6. Decimal Form: The exact form of a radical expression can be approximated by a decimal, which is often more useful for practical calculations. However, this form is an approximation and not exact.