> [!summary] Multiplying by the conjugate is a technique to simplify a complicated fraction. One that has a square root in the denominator or numerator. >[!info]+ Read Time **⏱ 2 mins** # Definition Multiplying by the conjugate is an algebraic technique used on functions that are in the form $\frac{A}{B}$ where either $A$ or $B$ are in the form $a\pm b$ where $a$ or $b$ is a square root ($\sqrt{ a }$) Often in multiplying by the conjugate, it is because the evaluation of a [[Limits|limit]] is in the form $\frac{0}{0}\space \text{or} \space \frac{\infty}{\infty}$, an [[Indeterminate Forms|indeterminate form]]. This is saying that as you approach some value, it becomes undefined. This technique creates a new function, identical to the original, upset that it estimates the value at the point where it is undefined. The estimation is done by exploiting the fact that $(a+b)(a-b)= a^2-b^2$ to reduce the square root part of the function into a normal value. In doing so, you create a replica of the original function, and an estimation of what the value is most likely to be at the point where it was undefined. ## Technique A function in the form $\frac{A}{B}$ where either $A,B$ are in the form $a\pm b$ and a or b is a square root function. As well, the limit of the function is an indeterminate form, then the following must be done to simplify it. 1. Find the part of the function in the form $\sqrt{ a }\pm b$ 2. Multiplying the top and bottom of the function by the same function with opposite signs (e.g. multiplying the top and bottom of the function $\sqrt{ a }- b$ if the function was in the form $\sqrt{ a }+b$) 3. Simplify the top and bottom Step 2 is called multiplying by the conjugate (opposite)