Simplify ( sixth root of x^13y^5)/( square root of xy)

Solution:

Step:1

Simplify the numerator.

Step:1.1

Express $x^{13} y^{5}$ as $(x^2)^6 \cdot xy^5$.

Step:1.1.1

Extract $x^{12}$ as a factor: $\frac{\sqrt[6]{x^{12} \cdot xy^5}}{\sqrt{xy}}$.

Step:1.1.2

Represent $x^{12}$ as $(x^2)^6$: $\frac{\sqrt[6]{(x^2)^6 \cdot xy^5}}{\sqrt{xy}}$.

Step:1.1.3

Enclose with brackets: $\frac{\sqrt[6]{((x^2)^6 \cdot (xy^5))}}{\sqrt{xy}}$.

Step:1.2

Extract terms from under the sixth root: $\frac{x^2 \sqrt[6]{xy^5}}{\sqrt{xy}}$.

Step:2

Multiply $\frac{x^2 \sqrt[6]{xy^5}}{\sqrt{xy}}$ by $\frac{\sqrt{xy}}{\sqrt{xy}}$.

Step:3

Simplify the terms.

Step:3.1

Simplify the denominator.

Step:3.1.1

Multiply $\frac{x^2 \sqrt[6]{xy^5} \sqrt{xy}}{\sqrt{xy} \sqrt{xy}}$.

Step:3.1.2

Express $\sqrt{xy}$ as $(xy)^{\frac{1}{2}}$.

Step:3.1.3

Raise $(xy)^{\frac{1}{2}}$ to the power of 2: $\frac{x^2 \sqrt[6]{xy^5} \sqrt{xy}}{((xy)^{\frac{1}{2}})^2}$.

Step:3.1.4

Apply the exponent rule: $\frac{x^2 \sqrt[6]{xy^5} \sqrt{xy}}{(xy)^{1}}$.

Step:3.1.5

Simplify the expression: $\frac{x^2 \sqrt[6]{xy^5} \sqrt{xy}}{xy}$.

Step:3.2

Cancel the common $x$ factor.

Step:3.2.1

Factor $x$ from the numerator: $\frac{x(x \sqrt[6]{xy^5} \sqrt{xy})}{xy}$.

Step:3.2.2

Cancel the common $x$ factor: $\frac{x \sqrt[6]{xy^5} \sqrt{xy}}{y}$.

Step:4

Simplify the numerator.

Step:4.1

Use the least common index of 6 for rewriting.

Step:4.1.1

Express $\sqrt{xy}$ as $(xy)^{\frac{1}{2}}$: $\frac{x((xy)^{\frac{1}{2}} \sqrt[6]{xy^5})}{y}$.

Step:4.1.2

Rewrite $(xy)^{\frac{1}{2}}$ as $\sqrt[6]{(xy)^3}$.

Step:4.2

Combine under a single sixth root: $\frac{x \sqrt[6]{(xy)^3 \cdot xy^5}}{y}$.

Step:4.3

Apply the product rule to $xy$: $\frac{x \sqrt[6]{x^3y^3 \cdot xy^5}}{y}$.

Step:4.4

Combine exponents: $\frac{x \sqrt[6]{x^4y^8}}{y}$.

Step:5

Simplify the numerator further.

Step:5.1

Express $x^4y^8$ as $y^6(x^4y^2)$.

Step:5.2

Extract $y^6$ from under the radical: $\frac{xy \sqrt[6]{x^4y^2}}{y}$.

Step:5.3

Rewrite $x^4y^2$ as $(x^2y)^2$.

Step:5.4

Express the sixth root as a cube root of a square root: $\frac{xy \sqrt[3]{\sqrt{(x^2y)^2}}}{y}$.

Step:5.5

Assuming positive real numbers, simplify to: $\frac{xy \sqrt[3]{x^2y}}{y}$.

Step:6

Cancel the common $y$ factor: $x \sqrt[3]{x^2y}$.

Knowledge Notes:

1. Radical Simplification: Simplifying expressions involving roots (square roots, cube roots, etc.) often involves factoring out perfect powers and pulling them out of the radical.

2. Exponent Rules: The power rule states that $a^m \cdot a^n = a^{m+n}$. This is used to combine like terms under a radical or when raising an expression to a power.

3. Rational Exponents: A root can be expressed as a rational exponent, such that $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. This is useful for simplifying expressions with different types of roots.

4. Combining Radicals: When radicals have the same index, they can be combined under a single radical sign using the product rule for radicals: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

5. Simplifying Complex Fractions: When simplifying complex fractions, it's often helpful to multiply the numerator and denominator by a common factor to eliminate the radical in the denominator.

6. Cancelling Common Factors: When a factor appears in both the numerator and denominator, it can be cancelled out to simplify the expression.

7. Assumption of Positive Real Numbers: When simplifying radicals, it is often assumed that the variables represent positive real numbers to avoid complex numbers and ensure that the roots are real.