Simplify 1/(3 square root of x-6)

Solution:

Step 1:

Rationalize the denominator of $\frac{1}{3\sqrt{x-6}}$ by multiplying both numerator and denominator by $\sqrt{x-6}$:
$\frac{1}{3\sqrt{x-6}}\times \frac{\sqrt{x-6}}{\sqrt{x-6}}$

Step 2:

Simplify the expression by performing the multiplication.

Step 2.1:

Multiply the numerator by $\sqrt{x-6}$:
$\frac{\sqrt{x-6}}{3\sqrt{x-6}\sqrt{x-6}}$

Step 2.2:

Combine the square root terms in the denominator.

Step 2.3:

Express $\sqrt{x-6}$ as a power:
$\frac{\sqrt{x-6}}{3(\sqrt{x-6}{)}^1\sqrt{x-6}}$

Step 2.4:

Combine the like terms using exponent rules.

Step 2.5:

Apply the exponent rule $a^m\cdot a^n=a^{m+n}$:
$\frac{\sqrt{x-6}}{3(\sqrt{x-6}{)}^{1+1}}$

Step 2.6:

Add the exponents:
$\frac{\sqrt{x-6}}{3(\sqrt{x-6}{)}^2}$

Step 2.7:

Simplify the square of the square root.

Step 2.7.1:

Rewrite the square root as a fractional exponent:
$\frac{\sqrt{x-6}}{3((x-6{)}^{\frac{1}{2}}{)}^2}$

Step 2.7.2:

Apply the power of a power rule:
$\frac{\sqrt{x-6}}{3(x-6{)}^{\frac{1}{2}\cdot 2}}$

Step 2.7.3:

Multiply the exponents.

Step 2.7.4:

Simplify the exponent.

Step 2.7.4.1:

Cancel out the common factors in the exponent:
$\frac{\sqrt{x-6}}{3(x-6{)}^{\frac{\cancel{2}}{\cancel{2}}}}$

Step 2.7.4.2:

Rewrite the simplified expression:
$\frac{\sqrt{x-6}}{3(x-6{)}^1}$

Step 2.7.5:

Final simplification:
$\frac{\sqrt{x-6}}{3(x-6)}$

Knowledge Notes:

To simplify the expression $\frac{1}{3\sqrt{x-6}}$, we use the process of rationalizing the denominator. This involves getting rid of the square root in the denominator by multiplying the expression by a form of one that contains the square root, which in this case is $\frac{\sqrt{x-6}}{\sqrt{x-6}}$.

The steps include:

1. Multiplication of the numerator and denominator by the conjugate of the denominator to eliminate the square root.

2. Application of the distributive property (if necessary) to multiply terms.

3. Simplification of the resulting expression by combining like terms and using exponent rules.

4. The power rule for exponents, which states that $a^m\cdot a^n=a^{m+n}$, is used to combine terms with the same base.

5. The power of a power rule, which states that $(a^m{)}^n=a^{mn}$, is used to simplify expressions with powers raised to another power.

6. Simplification of the expression by canceling out common factors and reducing the expression to its simplest form.

These rules and properties are fundamental to algebra and are used extensively in simplifying expressions and solving equations.