Simplify ( cube root of 4ab^5)/(4ab^2)

Solution:

Step 1: Simplify the numerator.

• Step 1.1: Express $4a{b}^5$ as a product involving $b^3$.

  • Step 1.1.1: Extract $b^3$ from the expression. $\frac{\sqrt[3]{4a{b}^3\cdot b^2}}{4a{b}^2}$
  • Step 1.1.2: Rearrange to place $a$ next to $4$. $\frac{\sqrt[3]{4b^{3}a{b}^2}}{4a{b}^2}$
  • Step 1.1.3: Switch the positions of $4$ and $b^3$. $\frac{\sqrt[3]{b^3\cdot 4a{b}^2}}{4a{b}^2}$
  • Step 1.1.4: Enclose $4a{b}^2$ in parentheses. $\frac{\sqrt[3]{b^3\cdot (4a{b}^2)}}{4a{b}^2}$
  • Step 1.1.5: Ensure parentheses are correctly placed. $\frac{\sqrt[3]{b^3\cdot (4a{b}^2)}}{4a{b}^2}$

• Step 1.2: Extract cube root terms from under the radical. $\frac{b\sqrt[3]{4a{b}^2}}{4a{b}^2}$

Step 2: Eliminate common factors.

• Step 2.1: Factor out $b$ from the numerator. $\frac{b(\sqrt[3]{4a{b}^2})}{4a{b}^2}$

• Step 2.2: Simplify by canceling common factors.

  • Step 2.2.1: Factor $b$ from the denominator. $\frac{b(\sqrt[3]{4a{b}^2})}{b(4ab)}$
  • Step 2.2.2: Cancel out the common $b$ term. $\frac{\cancel{b}\sqrt[3]{4a{b}^2}}{\cancel{b}(4ab)}$
  • Step 2.2.3: Write the simplified expression. $\frac{\sqrt[3]{4a{b}^2}}{4ab}$

Knowledge Notes:

To simplify the given expression $\frac{\sqrt[3]{4a{b}^5}}{4a{b}^2}$, we need to understand several mathematical concepts:

1. Cube Root: The cube root of a number $x$, denoted as $\sqrt[3]{x}$, is a value that, when multiplied by itself three times, gives $x$. For instance, $\sqrt[3]{8}=2$ because $2\times 2\times 2=8$.

2. Factoring: This involves breaking down a number or expression into a product of its factors. For example, $4a{b}^5$ can be factored into $b^3\cdot (4a{b}^2)$ because $b^3\cdot b^2=b^5$.

3. Simplifying Radicals: When simplifying expressions involving radicals, any factor inside the radical that is a perfect power of the index (in this case, a cube) can be taken out of the radical. For example, $\sqrt[3]{b^3}$ simplifies to $b$.

4. Cancellation: In fractions, if the same factor appears in both the numerator and the denominator, it can be canceled out. For instance, in $\frac{b(4a{b}^2)}{b(4ab)}$, the $b$ in the numerator and denominator can be canceled.

5. Algebraic Manipulation: This involves rearranging terms, factoring, and simplifying expressions to reach a more simplified or desired form.

By applying these concepts, we can simplify the given expression by extracting cube roots, factoring out common terms, and canceling them to reach the final simplified form.