Simplify 9/( square root of x-9)

Solution:

Step:1

Rationalize the denominator by multiplying $\frac{9}{\sqrt{x-9}}$ with $\frac{\sqrt{x-9}}{\sqrt{x-9}}$ to get $\frac{9}{\sqrt{x-9}}\cdot \frac{\sqrt{x-9}}{\sqrt{x-9}}$.

Step:2

Proceed to simplify the expression in the denominator.

Step:2.1

Multiply numerator and denominator by $\sqrt{x-9}$ to obtain $\frac{9\sqrt{x-9}}{\sqrt{x-9}\cdot \sqrt{x-9}}$.

Step:2.2

Recognize that $\sqrt{x-9}$ is raised to the power of $1$, giving us $\frac{9\sqrt{x-9}}{(\sqrt{x-9}{)}^1\cdot \sqrt{x-9}}$.

Step:2.3

Reiterate the exponent of $1$ for clarity, resulting in $\frac{9\sqrt{x-9}}{(\sqrt{x-9}{)}^1\cdot (\sqrt{x-9}{)}^1}$.

Step:2.4

Apply the rule of exponents $a^m\cdot a^n=a^{m+n}$ to combine the exponents, yielding $\frac{9\sqrt{x-9}}{(\sqrt{x-9}{)}^{1+1}}$.

Step:2.5

Add the exponents $1$ and $1$ together to get $\frac{9\sqrt{x-9}}{(\sqrt{x-9}{)}^2}$.

Step:2.6

Express $(\sqrt{x-9}{)}^2$ as $x-9$.

Step:2.6.1

Rewrite $\sqrt{x-9}$ using the property $\sqrt[n]{a^x}=a^{\frac{x}{n}}$, resulting in $\frac{9\sqrt{x-9}}{((x-9{)}^{\frac{1}{2}}{)}^2}$.

Step:2.6.2

Utilize the power rule $(a^m{)}^n=a^{m\cdot n}$ to simplify the expression to $\frac{9\sqrt{x-9}}{(x-9{)}^{\frac{1}{2}\cdot 2}}$.

Step:2.6.3

Combine the exponents $\frac{1}{2}$ and $2$ to get $\frac{9\sqrt{x-9}}{(x-9{)}^{\frac{2}{2}}}$.

Step:2.6.4

Eliminate the common factor of $2$ in the exponent.

Step:2.6.4.1

Cancel out the common factors to simplify to $\frac{9\sqrt{x-9}}{(x-9{)}^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step:2.6.4.2

Rewrite the expression as $\frac{9\sqrt{x-9}}{(x-9{)}^1}$.

Step:2.6.5

Simplify the expression to $\frac{9\sqrt{x-9}}{x-9}$.

Knowledge Notes:

The problem involves simplifying a rational expression with a radical in the denominator. The goal is to eliminate the radical to make the expression easier to work with, especially when performing further operations like addition, subtraction, or comparison.

Key knowledge points include:

1. Rationalizing the Denominator: This is the process of eliminating radicals from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by an appropriate form of 1 that contains the radical, which in this case is $\frac{\sqrt{x-9}}{\sqrt{x-9}}$.

2. Simplifying Radicals: When a radical is raised to a power that matches the index of the radical, the radical sign can be removed. For example, $(\sqrt{x}{)}^2=x$.

3. Exponent Rules: Several rules for exponents are used in the solution, including:

   • Product Rule: $a^m\cdot a^n=a^{m+n}$, which allows us to combine exponents when multiplying like bases.

   • Power of a Power Rule: $(a^m{)}^n=a^{m\cdot n}$, which allows us to multiply exponents when raising a power to another power.

4. Simplifying Expressions: After applying the exponent rules, we simplify the expression by canceling out common factors in the exponent, leading to a simplified form of the original expression.

These concepts are fundamental in algebra and are widely applicable in various areas of mathematics, including calculus, number theory, and mathematical modeling.