Simplify ( cube root of 27xy^7)/( cube root of xy)
The problem asks for the simplification of a fraction that consists of cube roots in both its numerator and denominator. Specifically, the problem is to simplify the algebraic expression formed by taking the cube root of 27xy^7 and dividing it by the cube root of xy. The solution will involve applying the properties of roots and exponents to simplify the expression to its simplest form.
$\frac{\sqrt[3]{27 x y^{7}}}{\sqrt[3]{x y}}$
Answer
Solution:
Step 1:
Merge the cube roots $\sqrt[3]{27xy^7}$ and $\sqrt[3]{xy}$ into a single cube root expression: $\sqrt[3]{\frac{27xy^7}{xy}}$.
Step 2:
Simplify the fraction $\frac{27xy^7}{xy}$ 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:
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 $3^3 = 27$.
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.
Simplifying Fractions: When simplifying fractions within a radical, we look for common factors in the numerator and denominator that can be cancelled out.
Properties of Exponents: When dividing terms with the same base, we subtract the exponents. For example, $y^7 / y = y^{7-1} = y^6$.
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.
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.
