Simplify square root of (z^6)/(27x^9)

Solution:

Step 1:

Express $\frac{z^6}{27x^9}$ as ${\left(\frac{z^3}{3x^4}\right)}^2\times \frac{1}{3x}$.

Step 1.1:

Extract the square of the perfect cube $z^3$ from $z^6$ as $\sqrt{\frac{(z^3{)}^2\times 1}{27x^9}}$.

Step 1.2:

Extract the square of the perfect cube $3x^4$ from $27x^9$ as $\sqrt{\frac{(z^3{)}^2\times 1}{(3x^4{)}^2\times 3x}}$.

Step 1.3:

Reorganize the fraction to $\sqrt{{\left(\frac{z^3}{3x^4}\right)}^2\times \frac{1}{3x}}$.

Step 2:

Extract the terms from under the square root as $\frac{z^3}{3x^4}\times \sqrt{\frac{1}{3x}}$.

Step 3:

Rewrite the square root as a fraction $\frac{\sqrt{1}}{\sqrt{3x}}$.

Step 4:

Combine the expressions as $\frac{z^3\times \sqrt{1}}{3x^4\times \sqrt{3x}}$.

Step 5:

Recognize that the square root of $1$ is $1$.

Step 6:

Multiply $z^3$ by $1$ to get $\frac{z^3}{3x^4\times \sqrt{3x}}$.

Step 7:

Multiply by $\frac{\sqrt{3x}}{\sqrt{3x}}$ to rationalize the denominator.

Step 8:

Simplify the denominator.

Step 8.1:

Multiply $\frac{z^3\times \sqrt{3x}}{3x^4\times \sqrt{3x}\times \sqrt{3x}}$.

Step 8.2:

Relocate $\sqrt{3x}$.

Step 8.3:

Raise $\sqrt{3x}$ to the power of $1$.

Step 8.4:

Raise $\sqrt{3x}$ to the power of $1$ again.

Step 8.5:

Apply the power rule $a^m\times a^n=a^{m+n}$ to combine exponents.

Step 8.6:

Sum the exponents $1$ and $1$.

Step 8.7:

Rewrite $(\sqrt{3x}{)}^2$ as $3x$.

Step 8.7.1:

Express $\sqrt{3x}$ as $(3x{)}^{\frac{1}{2}}$.

Step 8.7.2:

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

Step 8.7.3:

Combine $\frac{1}{2}$ and $2$.

Step 8.7.4:

Cancel out the common factor of $2$.

Step 8.7.5:

Simplify the expression.

Step 9:

Add the exponents of $x^4$ and $x$.

Step 9.1:

Relocate $x$.

Step 9.2:

Multiply $x$ by $x^4$.

Step 9.2.1:

Raise $x$ to the power of $1$.

Step 9.2.2:

Combine the exponents using the power rule $a^m\times a^n=a^{m+n}$.

Step 9.3:

Add the exponents $1$ and $4$.

Step 10:

Multiply $3$ by $3$ to get the final result $\frac{z^3\times \sqrt{3x}}{9x^5}$.

Knowledge Notes:

To simplify the square root of a fraction, we can use the following steps and knowledge points:

1. Factorization: Break down the numerator and the denominator into factors that can help simplify the expression.

2. Square Roots: Recognize that the square root of a square number or variable is the base of that square. For example, $\sqrt{x^2}=x$.

3. Rationalizing the Denominator: Multiply the numerator and the denominator by a term that will eliminate the square root in the denominator.

4. Fraction Rules: Understand that $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ and use this to separate terms under the square root.

5. Power Rules: Use the rules of exponents to simplify expressions, such as $a^m\times a^n=a^{m+n}$ and $(a^m{)}^n=a^{mn}$.

6. Simplification: Combine like terms and reduce fractions to their simplest form.

In this problem, we applied these principles to simplify the square root of a fraction involving powers of variables and numbers. The process involved extracting perfect squares, rationalizing the denominator, and simplifying the expression using exponent rules.