Simplify cube root of (24x^3)/(36x)

Solution:

Step 1: Simplify the Fraction Inside the Cube Root

Divide the numerator and denominator by their greatest common factor.

Step 1.1

Extract the factor of 12 from the numerator: $\sqrt[3]{\frac{12(2x^3)}{36x}}$

Step 1.2

Extract the factor of 12 from the denominator: $\sqrt[3]{\frac{12(2x^3)}{12(3x)}}$

Step 1.3

Eliminate the common factor of 12: $\sqrt[3]{\frac{\cancel{12}(2x^3)}{\cancel{12}(3x)}}$

Step 1.4

Rewrite the simplified fraction: $\sqrt[3]{\frac{2x^3}{3x}}$

Step 2: Further Simplify the Fraction

Remove common factors of $x$ from the numerator and denominator.

Step 2.1

Factor out $x$ from the numerator: $\sqrt[3]{\frac{x(2x^2)}{3x}}$

Step 2.2: Cancel Common Factors

Step 2.2.1

Factor out $x$ from the denominator: $\sqrt[3]{\frac{x(2x^2)}{x \cdot 3}}$

Step 2.2.2

Cancel the $x$: $\sqrt[3]{\frac{\cancel{x}(2x^2)}{\cancel{x} \cdot 3}}$

Step 2.2.3

Rewrite the expression: $\sqrt[3]{\frac{2x^2}{3}}$

Step 3: Separate the Cube Root

Split the cube root of the fraction into the cube root of the numerator and the cube root of the denominator: $\frac{\sqrt[3]{2x^2}}{\sqrt[3]{3}}$

Step 4: Rationalize the Denominator

Multiply by the conjugate of the cube root of the denominator to rationalize it.

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

Step 5: Simplify the Denominator

Combine and simplify the terms in the denominator.

Step 5.1

Multiply the numerator and the denominator by $\left(\sqrt[3]{3}\right)^2$: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{\sqrt[3]{3}\left(\sqrt[3]{3}\right)^2}$

Step 5.2

Raise $\sqrt[3]{3}$ to the first power: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{\left(\sqrt[3]{3}\right)^1\left(\sqrt[3]{3}\right)^2}$

Step 5.3

Combine exponents using the power rule $a^m a^n = a^{m+n}$: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{\left(\sqrt[3]{3}\right)^{1+2}}$

Step 5.4

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

Step 5.5: Rewrite the Cube Root as an Exponent

Step 5.5.1

Express $\sqrt[3]{3}$ as $3^{\frac{1}{3}}$: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{\left(3^{\frac{1}{3}}\right)^3}$

Step 5.5.2

Apply the power rule $\left(a^m\right)^n = a^{mn}$: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{3^{\frac{1}{3} \cdot 3}}$

Step 5.5.3

Multiply the exponents: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{3^{\frac{3}{3}}}$

Step 5.5.4

Simplify the exponent: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{3^1}$

Step 5.5.5

Evaluate the exponent: $\frac{\sqrt[3]{2x^2}\left(\sqrt[3]{3}\right)^2}{3}$

Step 6: Simplify the Numerator

Combine the cube roots in the numerator.

Step 6.1

Rewrite $\left(\sqrt[3]{3}\right)^2$ as $\sqrt[3]{3^2}$: $\frac{\sqrt[3]{2x^2}\sqrt[3]{3^2}}{3}$

Step 6.2

Calculate $3^2$: $\frac{\sqrt[3]{2x^2}\sqrt[3]{9}}{3}$

Step 7: Final Simplification

Combine the cube roots into a single cube root.

Step 7.1

Use the product rule for radicals: $\frac{\sqrt[3]{2x^2 \cdot 9}}{3}$

Step 7.2

Multiply $2$ by $9$: $\frac{\sqrt[3]{18x^2}}{3}$

Knowledge Notes:

The problem-solving process involves simplifying a cube root of a fraction, which requires understanding of several mathematical concepts:

1. Factorization: This is the process of breaking down numbers into their prime factors or breaking down algebraic expressions into simpler components that can be canceled or combined.

2. Radicals and Rational Exponents: Radicals (such as cube roots) can also be expressed as rational exponents. For example, $\sqrt[3]{a} = a^{\frac{1}{3}}$. This property is used to simplify expressions involving cube roots.

3. Simplifying Fractions: When simplifying fractions, any common factors in the numerator and denominator can be canceled out.

4. Properties of Exponents: The power rule $a^m a^n = a^{m+n}$ and the rule $\left(a^m\right)^n = a^{mn}$ are used to simplify expressions with exponents.

5. Rationalizing the Denominator: When a radical is present in the denominator, it is often desirable to "rationalize" the denominator by eliminating the radical. This is done by multiplying the numerator and denominator by a suitable form of 1 that contains the radical.

6. Product Rule for Radicals: The product rule states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$, which allows us to combine radicals under a single radical sign.

By applying these concepts in a step-by-step manner, the given cube root expression is simplified to a more manageable form.