Simplify (3k^3)/( square root of 18k)

Solution:

Step 1: Simplify the square root in the denominator.

Step 1.1: Express $18k$ as a product of its prime factors.

$18k = 3^2 \cdot 2k$

Step 1.1.1: Extract the square factor from $18$.

$\frac{3k^3}{\sqrt{9 \cdot 2k}}$

Step 1.1.2: Represent $9$ as $3^2$.

$\frac{3k^3}{\sqrt{3^2 \cdot 2k}}$

Step 1.1.3: Enclose the terms under the radical in parentheses.

$\frac{3k^3}{\sqrt{3^2 \cdot (2k)}}$

Step 1.2: Extract the square root of the perfect square.

$\frac{3k^3}{3\sqrt{2k}}$

Step 2: Eliminate the common factor of $3$.

Step 2.1: Reduce the fraction by canceling out the common factor.

$\frac{\cancel{3}k^3}{\cancel{3}\sqrt{2k}}$

Step 2.2: Present the simplified expression.

$\frac{k^3}{\sqrt{2k}}$

Step 3: Rationalize the denominator by multiplying by the conjugate.

$\frac{k^3}{\sqrt{2k}} \cdot \frac{\sqrt{2k}}{\sqrt{2k}}$

Step 4: Simplify the expression in the denominator.

Step 4.1: Multiply the numerator and denominator by $\sqrt{2k}$.

$\frac{k^3\sqrt{2k}}{\sqrt{2k}\sqrt{2k}}$

Step 4.2: Raise the square root to the first power.

$\frac{k^3\sqrt{2k}}{(\sqrt{2k})^1\sqrt{2k}}$

Step 4.3: Repeat the exponentiation for clarity.

$\frac{k^3\sqrt{2k}}{(\sqrt{2k})^1(\sqrt{2k})^1}$

Step 4.4: Apply the exponent rule $a^m a^n = a^{m+n}$.

$\frac{k^3\sqrt{2k}}{(\sqrt{2k})^{1+1}}$

Step 4.5: Sum the exponents.

$\frac{k^3\sqrt{2k}}{(\sqrt{2k})^2}$

Step 4.6: Convert the square of the square root back to the original number.

$\frac{k^3\sqrt{2k}}{(2k)^1}$

Step 5: Cancel the common $k$ terms.

Step 5.1: Factor $k$ from $k^3\sqrt{2k}$.

$\frac{k(k^2\sqrt{2k})}{2k}$

Step 5.2: Reduce the fraction by canceling out the common $k$.

$\frac{\cancel{k}(k^2\sqrt{2k})}{\cancel{k}\cdot 2}$

Step 5.3: Present the final simplified expression.

$\frac{k^2\sqrt{2k}}{2}$

Knowledge Notes:

1. Square Roots and Radicals: The square root of a number $a$ is a value that, when multiplied by itself, gives $a$. Radicals are symbols used to denote roots, such as $\sqrt{\cdot}$ for square roots.

2. Prime Factorization: Breaking down a number into its prime factors helps simplify radical expressions.

3. Rationalizing the Denominator: This process involves removing the radical from the denominator of a fraction by multiplying both the numerator and the denominator by an appropriate form of 1, such as $\frac{\sqrt{b}}{\sqrt{b}}$.

4. Exponent Rules: The power rule states that $a^m \cdot a^n = a^{m+n}$. When raising a power to a power, you multiply the exponents, as in $(a^m)^n = a^{m \cdot n}$.

5. Simplifying Expressions: This involves reducing fractions to their simplest form by canceling out common factors in the numerator and denominator.

6. Algebraic Manipulation: The process of rearranging and simplifying algebraic expressions using various algebraic rules and properties.