Simplify ( square root of 9x+h- square root of 9x)/h

Solution:

Step 1:

Express the number 9 as the square of 3. $\frac{\sqrt{9x+h}-\sqrt{3^{2}x}}{h}$

Step 2:

Extract the square root of the perfect square from under the radical sign. $\frac{\sqrt{9x+h}-3\sqrt{x}}{h}$

Step 3:

Simplify the expression by recognizing that $3^2$ is the square of 3. $\frac{\sqrt{9x+h}-3\sqrt{x}}{h}$

Knowledge Notes:

To simplify the expression $\frac{\sqrt{9x+h}-\sqrt{9x}}{h}$, we need to apply several mathematical concepts:

1. Square Roots and Radicals: The square root of a number is a value that, when multiplied by itself, gives the original number. The radical symbol $\sqrt{}$ is used to denote the square root.

2. Simplifying Square Roots: When simplifying square roots, we look for perfect squares that can be taken out from under the radical. For example, $\sqrt{9x}$ can be simplified to $3\sqrt{x}$ because 9 is a perfect square and equals $3^2$.

3. Algebraic Manipulation: The expression involves algebraic manipulation, such as combining like terms and simplifying fractions.

By recognizing that 9 is a perfect square and can be written as $3^2$, we can simplify the square root of $9x$ to $3\sqrt{x}$. This step is crucial in simplifying the given expression.