Simplify ( square root of 9x+h- square root of 9x)/h
The problem provided is an algebraic expression that needs to be simplified. It involves a difference of square roots in the numerator and a single variable h in the denominator. The expression is a fraction and represents a form of the difference quotient which is a fundamental concept in calculus for defining the derivative of a function. The task is to simplify this expression by performing the appropriate algebraic manipulations, potentially involving rationalizing the numerator or combining like terms to reach a more simplified form of the expression. The goal is to find an equivalent expression that has been reduced to its lowest terms.
$\frac{\sqrt{9 x + h} - \sqrt{9 x}}{h}$
Answer
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:
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.
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 $3^2$.
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.
