Simplify ( cube root of 120x^15)/( cube root of 5x)

Solution:

Step 1:

Merge the cube roots $\sqrt[3]{120x^{15}}$ and $\sqrt[3]{5x}$ into a single cube root expression: $\sqrt[3]{\frac{120x^{15}}{5x}}$.

Step 2:

Simplify the fraction within the cube root by removing common factors.

Step 2.1:

Extract the factor of $5$ from the numerator: $\sqrt[3]{\frac{5(24x^{15})}{5x}}$.

Step 2.2:

Extract the factor of $5$ from the denominator: $\sqrt[3]{\frac{5(24x^{15})}{5(x)}}$.

Step 2.3:

Eliminate the matching factor of $5$: $\sqrt[3]{\frac{\cancel{5}(24x^{15})}{\cancel{5}x}}$.

Step 2.4:

Reformulate the expression: $\sqrt[3]{\frac{24x^{15}}{x}}$.

Step 3:

Further simplify by canceling out the common $x$ terms.

Step 3.1:

Factor out $x$ from the numerator: $\sqrt[3]{\frac{x(24x^{14})}{x}}$.

Step 3.2:

Proceed with the cancellation of common factors.

Step 3.2.1:

Express $x$ as $x^1$: $\sqrt[3]{\frac{x(24x^{14})}{x^1}}$.

Step 3.2.2:

Factor $x$ from $x^1$: $\sqrt[3]{\frac{x(24x^{14})}{x\cdot 1}}$.

Step 3.2.3:

Cancel the matching $x$ factor: $\sqrt[3]{\frac{\cancel{x}(24x^{14})}{\cancel{x}\cdot 1}}$.

Step 3.2.4:

Rephrase the expression: $\sqrt[3]{\frac{24x^{14}}{1}}$.

Step 3.2.5:

Divide $24x^{14}$ by $1$: $\sqrt[3]{24x^{14}}$.

Step 4:

Decompose $24x^{14}$ into factors that are perfect cubes and those that are not.

Step 4.1:

Factor out $8$ from $24$: $\sqrt[3]{8(3)x^{14}}$.

Step 4.2:

Represent $8$ as $2^3$: $\sqrt[3]{2^3\cdot 3x^{14}}$.

Step 4.3:

Isolate $x^{12}$: $\sqrt[3]{2^3\cdot 3(x^{12}{x}^2)}$.

Step 4.4:

Express $x^{12}$ as $(x^4{)}^3$: $\sqrt[3]{2^3\cdot 3((x^4{)}^3{x}^2)}$.

Step 4.5:

Rearrange $3$: $\sqrt[3]{2^3((x^4{)}^3)\cdot 3x^2}$.

Step 4.6:

Combine $2^3$ and $(x^4{)}^3$ into a single cube: $\sqrt[3]{(2x^4{)}^3\cdot 3x^2}$.

Step 4.7:

Enclose the terms with parentheses: $\sqrt[3]{((2x^4{)}^3)\cdot (3x^2)}$.

Step 5:

Extract terms that are perfect cubes from under the cube root: $2x^4\sqrt[3]{3x^2}$.

Knowledge Notes:

To simplify a radical expression involving cube roots, we follow these steps:

1. Combine cube roots into a single cube root if possible.

2. Simplify the expression under the cube root by canceling common factors.

3. Factor the expression under the cube root into perfect cubes and non-perfect cubes.

4. Extract the perfect cubes from under the cube root.

5. Rewrite the expression with the extracted terms outside the cube root.

Key concepts used in this problem include:

• Cube root: The cube root of a number $a$ is a number $b$ such that $b^3=a$.

• Simplifying radicals: This involves finding and extracting perfect squares, cubes, etc., from under the radical sign.

• Factoring: The process of breaking down numbers into their constituent prime factors or other divisible components.

• Cancelling common factors: When a factor appears in both the numerator and denominator of a fraction, it can be canceled out.

In the context of this problem, we used the property that $\sqrt[3]{a^3}=a$ to extract perfect cubes from under the cube root. We also used the fact that $\sqrt[3]{ab}=\sqrt[3]{a}\sqrt[3]{b}$ to separate terms under the cube root.