Simplify ( cube root of 27xy^7)/( cube root of xy)

Solution:

Step 1:

Merge the cube roots $\sqrt[3]{27x{y}^7}$ and $\sqrt[3]{xy}$ into a single cube root expression: $\sqrt[3]{\frac{27x{y}^7}{xy}}$.

Step 2:

Simplify the fraction $\frac{27x{y}^7}{xy}$ by removing common factors.

Step 2.1:

Eliminate the shared $x$ term: $\sqrt[3]{\frac{27\cancel{x}{y}^7}{\cancel{x}y}}$.

Step 2.2:

Represent the simplified fraction: $\sqrt[3]{\frac{27y^7}{y}}$.

Step 3:

Further reduce the fraction by dividing powers of $y$.

Step 3.1:

Extract $y$ from the numerator: $\sqrt[3]{\frac{y(27y^6)}{y}}$.

Step 3.2:

Proceed to cancel out identical factors.

Step 3.2.1:

Express $y$ in the denominator as $y^1$: $\sqrt[3]{\frac{y(27y^6)}{y^1}}$.

Step 3.2.2:

Isolate $y$ in the denominator: $\sqrt[3]{\frac{y(27y^6)}{y\cdot 1}}$.

Step 3.2.3:

Remove the common $y$ term: $\sqrt[3]{\frac{\cancel{y}(27y^6)}{\cancel{y}\cdot 1}}$.

Step 3.2.4:

Update the fraction: $\sqrt[3]{\frac{27y^6}{1}}$.

Step 3.2.5:

Divide $27y^6$ by $1$: $\sqrt[3]{27y^6}$.

Step 4:

Express $27y^6$ as a cube: $\sqrt[3]{(3y^2{)}^3}$.

Step 5:

Extract terms from the cube root, assuming all numbers are real: $3y^2$.

Knowledge Notes:

The process of simplifying cube roots involves several mathematical concepts:

1. Cube Root: The cube root of a number $a$, denoted as $\sqrt[3]{a}$, is a value that, when raised to the power of three, gives $a$. For example, $\sqrt[3]{27}=3$ because $3^3=27$.

2. Radicals: A radical expression involves roots, such as square roots or cube roots. Simplifying a radical expression often involves combining radicals and reducing the expression to its simplest form.

3. Simplifying Fractions: When simplifying fractions within a radical, we look for common factors in the numerator and denominator that can be cancelled out.

4. Properties of Exponents: When dividing terms with the same base, we subtract the exponents. For example, $y^7/y=y^{7-1}=y^6$.

5. Rationalizing the Denominator: When a radical is in the denominator, we often multiply the numerator and denominator by a form of 1 that will eliminate the radical from the denominator. In this case, however, we are simplifying a cube root, so the process is slightly different and involves cancelling common factors.

6. Real Numbers: The assumption of real numbers is important when pulling terms out from under the radical. We assume that all variables represent real numbers to avoid complex numbers in the solution.