Simplify 12 root of (81x^4y^12)/(16c^8d^4)

Solution:

Step:1

Express $81x^4y^{12}$ as $(3xy^3)^4$. Evaluate $\sqrt[12]{\frac{(3xy^3)^4}{16c^8d^4}}$.

Step:2

Express $16c^8d^4$ as $(2c^2d)^4$. Evaluate $\sqrt[12]{\frac{(3xy^3)^4}{(2c^2d)^4}}$.

Step:3

Combine the terms under the radical sign as $(\frac{3xy^3}{2c^2d})^4$. Evaluate $\sqrt[12]{(\frac{3xy^3}{2c^2d})^4}$.

Step:4

Convert the 12th root to a cube root of a fourth root: $\sqrt[3]{\sqrt[4]{(\frac{3xy^3}{2c^2d})^4}}$.

Step:5

Extract terms from under the radical, assuming all variables represent positive real numbers: $\sqrt[3]{\frac{3xy^3}{2c^2d}}$.

Step:6

Separate the cube root in the numerator and denominator: $\frac{\sqrt[3]{3xy^3}}{\sqrt[3]{2c^2d}}$.

Step:7

Simplify the numerator.

Step:7.1

Reorganize $3xy^3$ as $y^3 \cdot (3x)$.

Step:7.1.1

Rearrange to $\frac{\sqrt[3]{3y^3x}}{\sqrt[3]{2c^2d}}$.

Step:7.1.2

Switch the order of $3$ and $y^3$: $\frac{\sqrt[3]{y^3 \cdot 3x}}{\sqrt[3]{2c^2d}}$.

Step:7.1.3

Enclose with parentheses: $\frac{\sqrt[3]{y^3 \cdot (3x)}}{\sqrt[3]{2c^2d}}$.

Step:7.2

Extract terms from under the cube root: $\frac{y\sqrt[3]{3x}}{\sqrt[3]{2c^2d}}$.

Step:8

Multiply by $\frac{(\sqrt[3]{2c^2d})^2}{(\sqrt[3]{2c^2d})^2}$: $\frac{y\sqrt[3]{3x}}{\sqrt[3]{2c^2d}} \cdot \frac{(\sqrt[3]{2c^2d})^2}{(\sqrt[3]{2c^2d})^2}$.

Step:9

Combine and simplify the denominator.

Step:9.1

Multiply: $\frac{y\sqrt[3]{3x}(\sqrt[3]{2c^2d})^2}{\sqrt[3]{2c^2d}(\sqrt[3]{2c^2d})^2}$.

Step:9.2

Raise $\sqrt[3]{2c^2d}$ to the power of $1$: $\frac{y\sqrt[3]{3x}(\sqrt[3]{2c^2d})^2}{(\sqrt[3]{2c^2d})^1(\sqrt[3]{2c^2d})^2}$.

Step:9.3

Apply the power rule $a^ma^n = a^{m+n}$: $\frac{y\sqrt[3]{3x}(\sqrt[3]{2c^2d})^2}{(\sqrt[3]{2c^2d})^{1+2}}$.

Step:9.4

Add the exponents: $\frac{y\sqrt[3]{3x}(\sqrt[3]{2c^2d})^2}{(\sqrt[3]{2c^2d})^3}$.

Step:9.5

Convert the cube root to its equivalent power: $\frac{y\sqrt[3]{3x}(\sqrt[3]{2c^2d})^2}{(2c^2d)^1}$.

Step:10

Simplify the numerator.

Step:10.1

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

Step:10.2

Apply the product rule to $2c^2d$: $\frac{y\sqrt[3]{3x}\sqrt[3]{(2c^2)^2d^2}}{2c^2d}$.

Step:10.3

Apply the product rule to $2c^2$: $\frac{y\sqrt[3]{3x}\sqrt[3]{4(c^2)^2d^2}}{2c^2d}$.

Step:10.4

Square $2$: $\frac{y\sqrt[3]{3x}\sqrt[3]{4c^4d^2}}{2c^2d}$.

Step:10.5

Combine the exponents in $c^4$.

Step:10.5.1

Apply the power rule: $\frac{y\sqrt[3]{3x}\sqrt[3]{4c^{4}d^2}}{2c^2d}$.

Step:10.5.2

Multiply the exponents: $\frac{y\sqrt[3]{3x}\sqrt[3]{4c^{4}d^2}}{2c^2d}$.

Step:11

Cancel the common factors of $c$.

Step:11.1

Factor $c$ out of the numerator: $\frac{c(y\sqrt[3]{12xcd^2})}{2c^2d}$.

Step:11.2

Cancel the common $c$ factors: $\frac{y\sqrt[3]{12xcd^2}}{2cd}$.

Knowledge Notes:

1. Radical Simplification: The process involves expressing numbers as powers of their prime factors and simplifying under the radical sign.

2. Cube and Fourth Roots: These are specific types of radicals where the index is 3 (cube root) or 4 (fourth root).

3. Rationalizing the Denominator: Multiplying by a form of one to eliminate radicals from the denominator.

4. Power Rule: For any real number $a$ and integers $m$ and $n$, $a^m \cdot a^n = a^{m+n}$.

5. Product Rule for Radicals: For any nonnegative real numbers $a$ and $b$, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

6. Cancellation: If a factor appears in both the numerator and denominator, it can be canceled out.

7. Assumption of Positive Real Numbers: When simplifying radicals, it is typically assumed that all variables represent positive real numbers to avoid dealing with complex numbers.