Simplify ( cube root of 6c^4d^14)/( cube root of 48c^-2d^2)
The question is asking for the simplification of a mathematical expression involving cube roots and variables with exponents. The expression has a numerator and a denominator, both under the cube root. The task is to perform cube root division with the given terms, simplifying the result by applying the properties of exponents and the rules for division of cube roots. It involves reducing the expression to its simplest form by cancelling out common factors in the numerator and the denominator, and by simplifying the exponents of variables c and d as per the laws of exponents.
$\frac{\sqrt[3]{6 c^{4} d^{14}}}{\sqrt[3]{48 c^{- 2} d^{2}}}$
Answer
Solution:
Simplification Process
Step 1: Combine Radicals
Merge the cube roots into a single cube root expression: $\sqrt[3]{\frac{6c^4d^{14}}{48c^{-2}d^2}}$.
Step 2: Simplify Fraction
Simplify the fraction within the cube root.
Step 2.1
Extract the factor of 6 from the numerator: $\sqrt[3]{\frac{6(c^4d^{14})}{48c^{-2}d^2}}$.
Step 2.2
Extract the factor of 6 from the denominator: $\sqrt[3]{\frac{6(c^4d^{14})}{6(8c^{-2}d^2)}}$.
Step 2.3
Eliminate the common factor of 6: $\sqrt[3]{\frac{\cancel{6}(c^4d^{14})}{\cancel{6}(8c^{-2}d^2)}}$.
Step 2.4
Present the simplified expression: $\sqrt[3]{\frac{c^4d^{14}}{8c^{-2}d^2}}$.
Step 3: Apply Negative Exponent Rule
Move $c^{-2}$ to the numerator: $\sqrt[3]{\frac{c^4d^{14}c^2}{8d^2}}$.
Step 4: Combine Like Bases
Combine the powers of $c$ by adding their exponents.
Step 4.1
Position $c^2$ with $c^4$: $\sqrt[3]{\frac{c^2c^4d^{14}}{8d^2}}$.
Step 4.2
Apply the power rule $a^ma^n = a^{m+n}$: $\sqrt[3]{\frac{c^{2+4}d^{14}}{8d^2}}$.
Step 4.3
Sum the exponents for $c$: $\sqrt[3]{\frac{c^6d^{14}}{8d^2}}$.
Step 5: Reduce Common Factors
Cancel out common $d$ factors.
Step 5.1
Factor out $d^2$ from the numerator: $\sqrt[3]{\frac{d^2(c^6d^{12})}{8d^2}}$.
Step 5.2: Cancel Common Factors
Eliminate the common $d^2$ term.
Step 5.2.1
Extract $d^2$ from the denominator: $\sqrt[3]{\frac{d^2(c^6d^{12})}{d^2 \cdot 8}}$.
Step 5.2.2
Cancel out $d^2$: $\sqrt[3]{\frac{\cancel{d^2}(c^6d^{12})}{\cancel{d^2} \cdot 8}}$.
Step 5.2.3
Display the reduced expression: $\sqrt[3]{\frac{c^6d^{12}}{8}}$.
Step 6: Express as a Cube
Rewrite $c^6d^{12}$ as $(c^2d^4)^3$: $\sqrt[3]{\frac{(c^2d^4)^3}{8}}$.
Step 7: Represent 8 as a Cube
Express 8 as $2^3$: $\sqrt[3]{\frac{(c^2d^4)^3}{2^3}}$.
Step 8: Simplify the Cube Root
Rewrite the expression as a cube: $\sqrt[3]{\left(\frac{c^2d^4}{2}\right)^3}$.
Step 9: Simplify the Radical
Extract terms from under the cube root: $\frac{c^2d^4}{2}$.
Knowledge Notes:
To simplify the given expression, we used several algebraic rules and properties:
Combining Radicals: Radicals with the same index can be combined by manipulating the expression under the radical sign.
Simplifying Fractions: Common factors in the numerator and denominator can be cancelled out to simplify the fraction.
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.
Power Rule: When multiplying like bases, we add the exponents: $a^m \cdot a^n = a^{m+n}$.
Factoring and Cancelling: Common factors in the numerator and denominator can be factored out and cancelled.
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.
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.
