Simplify ( cube root of 6c^4d^14)/( cube root of 48c^-2d^2)

Solution:

Simplification Process

Step 1: Combine Radicals

Merge the cube roots into a single cube root expression: $\sqrt[3]{\frac{6c^4{d}^{14}}{48c^{-2}{d}^2}}$.

Step 2: Simplify Fraction

Simplify the fraction within the cube root.

Step 2.1

Extract the factor of 6 from the numerator: $\sqrt[3]{\frac{6(c^4{d}^{14})}{48c^{-2}{d}^2}}$.

Step 2.2

Extract the factor of 6 from the denominator: $\sqrt[3]{\frac{6(c^4{d}^{14})}{6(8c^{-2}{d}^2)}}$.

Step 2.3

Eliminate the common factor of 6: $\sqrt[3]{\frac{\cancel{6}(c^4{d}^{14})}{\cancel{6}(8c^{-2}{d}^2)}}$.

Step 2.4

Present the simplified expression: $\sqrt[3]{\frac{c^4{d}^{14}}{8c^{-2}{d}^2}}$.

Step 3: Apply Negative Exponent Rule

Move $c^{-2}$ to the numerator: $\sqrt[3]{\frac{c^4{d}^{14}{c}^2}{8d^2}}$.

Step 4: Combine Like Bases

Combine the powers of $c$ by adding their exponents.

Step 4.1

Position $c^2$ with $c^4$: $\sqrt[3]{\frac{c^2{c}^4{d}^{14}}{8d^2}}$.

Step 4.2

Apply the power rule $a^m{a}^n=a^{m+n}$: $\sqrt[3]{\frac{c^{2+4}{d}^{14}}{8d^2}}$.

Step 4.3

Sum the exponents for $c$: $\sqrt[3]{\frac{c^6{d}^{14}}{8d^2}}$.

Step 5: Reduce Common Factors

Cancel out common $d$ factors.

Step 5.1

Factor out $d^2$ from the numerator: $\sqrt[3]{\frac{d^2(c^6{d}^{12})}{8d^2}}$.

Step 5.2: Cancel Common Factors

Eliminate the common $d^2$ term.

Step 5.2.1

Extract $d^2$ from the denominator: $\sqrt[3]{\frac{d^2(c^6{d}^{12})}{d^2\cdot 8}}$.

Step 5.2.2

Cancel out $d^2$: $\sqrt[3]{\frac{\cancel{d^2}(c^6{d}^{12})}{\cancel{d^2}\cdot 8}}$.

Step 5.2.3

Display the reduced expression: $\sqrt[3]{\frac{c^6{d}^{12}}{8}}$.

Step 6: Express as a Cube

Rewrite $c^6{d}^{12}$ as $(c^2{d}^4{)}^3$: $\sqrt[3]{\frac{(c^2{d}^4{)}^3}{8}}$.

Step 7: Represent 8 as a Cube

Express 8 as $2^3$: $\sqrt[3]{\frac{(c^2{d}^4{)}^3}{2^3}}$.

Step 8: Simplify the Cube Root

Rewrite the expression as a cube: $\sqrt[3]{{\left(\frac{c^2{d}^4}{2}\right)}^3}$.

Step 9: Simplify the Radical

Extract terms from under the cube root: $\frac{c^2{d}^4}{2}$.

Knowledge Notes:

To simplify the given expression, we used several algebraic rules and properties:

1. Combining Radicals: Radicals with the same index can be combined by manipulating the expression under the radical sign.

2. Simplifying Fractions: Common factors in the numerator and denominator can be cancelled out to simplify the fraction.

3. Negative Exponents: The rule $b^{-n}=\frac{1}{b^n}$ allows us to move factors with negative exponents from the denominator to the numerator or vice versa.

4. Power Rule: When multiplying like bases, we add the exponents: $a^m\cdot a^n=a^{m+n}$.

5. Factoring and Cancelling: Common factors in the numerator and denominator can be factored out and cancelled.

6. Cube Roots and Cubes: A cube root of a cube, $\sqrt[3]{a^3}$, simplifies to $a$. This is because the cube root and the cube are inverse operations.

7. Radical Simplification: When the radicand is a perfect cube, the cube root can be simplified by taking the cube root of each factor separately.

By applying these rules step by step, we simplified the given complex radical expression to a much simpler form.