Simplify (4x^8w)/( square root of 9xw^2)

Solution:

Step 1: Simplify the denominator

Step 1.1: Express $9xw^2$ as a square

Step 1.1.1: Represent $9$ as $3^2$.

$$\frac{4x^8w}{\sqrt{3^2xw^2}}$$

Step 1.1.2: Rearrange $x$.

$$\frac{4x^8w}{\sqrt{3^2w^2x}}$$

Step 1.1.3: Express $3^2w^2$ as $(3w)^2$.

$$\frac{4x^8w}{\sqrt{(3w)^2x}}$$

Step 1.2: Extract terms from the radical.

$$\frac{4x^8w}{3w\sqrt{x}}$$

Step 2: Eliminate the common $w$ factor

Step 2.1: Remove the common $w$.

$$\frac{4x^8\cancel{w}}{3\cancel{w}\sqrt{x}}$$

Step 2.2: Present the simplified expression.

$$\frac{4x^8}{3\sqrt{x}}$$

Step 3: Rationalize the denominator

Multiply $\frac{4x^8}{3\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$.

Step 4: Combine and simplify the denominator

Step 4.1: Multiply the terms.

$$\frac{4x^8\sqrt{x}}{3\sqrt{x}\sqrt{x}}$$

Step 4.2: Rearrange $\sqrt{x}$.

$$\frac{4x^8\sqrt{x}}{3(\sqrt{x}\sqrt{x})}$$

Step 4.3: Express $\sqrt{x}$ as a power.

$$\frac{4x^8\sqrt{x}}{3((\sqrt{x})^1\sqrt{x})}$$

Step 4.4: Repeat the expression of $\sqrt{x}$ as a power.

$$\frac{4x^8\sqrt{x}}{3((\sqrt{x})^1(\sqrt{x})^1)}$$

Step 4.5: Apply the power rule $a^ma^n = a^{m+n}$.

$$\frac{4x^8\sqrt{x}}{3(\sqrt{x})^{1+1}}$$

Step 4.6: Add the exponents.

$$\frac{4x^8\sqrt{x}}{3(\sqrt{x})^2}$$

Step 4.7: Convert $(\sqrt{x})^2$ back to $x$.

Step 4.7.1: Rewrite $\sqrt{x}$ using $x^{\frac{1}{2}}$.

$$\frac{4x^8\sqrt{x}}{3(x^{\frac{1}{2}})^2}$$

Step 4.7.2: Apply the power rule $(a^m)^n = a^{mn}$.

$$\frac{4x^8\sqrt{x}}{3x^{\frac{1}{2}\cdot2}}$$

Step 4.7.3: Simplify the exponent.

$$\frac{4x^8\sqrt{x}}{3x^{\frac{2}{2}}}$$

Step 4.7.4: Simplify the fraction in the exponent.

$$\frac{4x^8\sqrt{x}}{3x^1}$$

Step 4.7.5: Simplify the expression.

$$\frac{4x^8\sqrt{x}}{3x}$$

Step 5: Cancel the common $x$ factors

Step 5.1: Factor $x$ from $4x^8\sqrt{x}$.

$$\frac{x(4x^7\sqrt{x})}{3x}$$

Step 5.2: Remove common factors.

Step 5.2.1: Factor $x$ from $3x$.

$$\frac{x(4x^7\sqrt{x})}{x\cdot3}$$

Step 5.2.2: Cancel the common $x$.

$$\frac{\cancel{x}(4x^7\sqrt{x})}{\cancel{x}\cdot3}$$

Step 5.2.3: Present the final expression.

$$\frac{4x^7\sqrt{x}}{3}$$

Knowledge Notes:

1. Simplifying Radicals: To simplify radicals, one can express the terms under the radical as powers and pull out perfect squares, cubes, etc.

2. Rationalizing the Denominator: When a radical is present in the denominator, it is common practice to multiply the fraction by a form of 1 that will eliminate the radical in the denominator.

3. Power Rule: For any nonzero number $a$ and any integers $m$ and $n$, the power rule states that $a^ma^n = a^{m+n}$.

4. Exponent Laws: When raising a power to a power, you multiply the exponents, as shown by $(a^m)^n = a^{mn}$.

5. Cancelling Common Factors: When the same factor appears in both the numerator and denominator, it can be cancelled out to simplify the fraction.

6. Square Roots and Exponents: The square root of a number $x$ can be written as $x^{\frac{1}{2}}$, and $(\sqrt{x})^2$ simplifies to $x$.