Simplify square root of (z^6)/(27x^9)
Brief Explanation of the Problem:
This problem is asking for the simplification of a given radical expression, which is the square root of a fraction. The numerator of the fraction is z raised to the power of 6 (z^6), and the denominator is 27 multiplied by x raised to the power of 9 (27x^9). The task is to apply the properties of exponents and the rules of simplifying square roots to the algebraic expression in order to express it in its simplest form.
$\sqrt{\frac{z^{6}}{27 x^{9}}}$
Answer
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 $\sqrt{\frac{(z^{3})^{2}\times1}{27x^{9}}}$.
Step 1.2:
Extract the square of the perfect cube $3x^{4}$ from $27x^{9}$ as $\sqrt{\frac{(z^{3})^{2}\times1}{(3x^{4})^{2}\times3x}}$.
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:
Factorization: Break down the numerator and the denominator into factors that can help simplify the expression.
Square Roots: Recognize that the square root of a square number or variable is the base of that square. For example, $\sqrt{x^2} = x$.
Rationalizing the Denominator: Multiply the numerator and the denominator by a term that will eliminate the square root in the denominator.
Fraction Rules: Understand that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ and use this to separate terms under the square root.
Power Rules: Use the rules of exponents to simplify expressions, such as $a^{m}\times a^{n} = a^{m+n}$ and $(a^{m})^{n} = a^{mn}$.
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.
