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Hii
(√2-1)/(√2+1)=x+y√3
By rationalization
(√2-1)²/(√2²-1²)
(2+1+4)/(2-1)
7/1
=7
a+b√3=7
I hope this much will help you
(√2-1)/(√2+1)=x+y√3
By rationalization
(√2-1)²/(√2²-1²)
(2+1+4)/(2-1)
7/1
=7
a+b√3=7
I hope this much will help you
JoelJohnson:
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When we put (vertically) large expressions inside of parentheses (or brackets, or curly braces, etc.), the parentheses don't resize to fit the expression and instead remain relatively small. For instance,$$f(x) = \pi(\frac{\sqrt{x}}{x-1})$$comes out as
To automatically adjust the size of parentheses to fit the expression inside of them, we type\left(...\right). If we do this for our equation above, we get
We can use \left and \right for all sorts of things... parentheses (as we saw), brackets$\left[...\right]$, braces $\left\{...\right\}, absolute values$\left|...\right|$, and much more (norms, floor and ceiling functions, inner products, etc.).
Lists
To make a list, such as a sequence, we use \dots. For example, $a_0,a_1,\dots,a_n$ will give us
Sums
There are two basic ways to write out sums. First, we can use + and \cdots. An example of this way would be $a_1+a_2+\cdots+a_n$ This will give us Second, we could use summation notation, or \sum. Such an example is$\sum_{i=0}^n a_i$, giving Note the use of superscripts and subscripts to obtain the summation index.
Products
Again, there are two basic ways to display products. First, we can use \cdot and \cdots. An example is$n! = n\cdot(n-1)\cdots 2\cdot 1$, which of course gives The alternative is to use product notation with \prod. For instance, $n! = \prod_{k=1}^n k$, giving
Equalities and Inequalities
Inequalities
the commands >, <, \geq, \leq, and \neqgive us and espectively.
Aligning Equations
To align multiple equations, we use the align*environment. For example, we might type a system of equations as follows:
\begin{align*} ax + by &= 1 \\ cx + dy &= 2 \\ ex + fy &= 3. \end{align*}
To automatically adjust the size of parentheses to fit the expression inside of them, we type\left(...\right). If we do this for our equation above, we get
We can use \left and \right for all sorts of things... parentheses (as we saw), brackets$\left[...\right]$, braces $\left\{...\right\}, absolute values$\left|...\right|$, and much more (norms, floor and ceiling functions, inner products, etc.).
Lists
To make a list, such as a sequence, we use \dots. For example, $a_0,a_1,\dots,a_n$ will give us
Sums
There are two basic ways to write out sums. First, we can use + and \cdots. An example of this way would be $a_1+a_2+\cdots+a_n$ This will give us Second, we could use summation notation, or \sum. Such an example is$\sum_{i=0}^n a_i$, giving Note the use of superscripts and subscripts to obtain the summation index.
Products
Again, there are two basic ways to display products. First, we can use \cdot and \cdots. An example is$n! = n\cdot(n-1)\cdots 2\cdot 1$, which of course gives The alternative is to use product notation with \prod. For instance, $n! = \prod_{k=1}^n k$, giving
Equalities and Inequalities
Inequalities
the commands >, <, \geq, \leq, and \neqgive us and espectively.
Aligning Equations
To align multiple equations, we use the align*environment. For example, we might type a system of equations as follows:
\begin{align*} ax + by &= 1 \\ cx + dy &= 2 \\ ex + fy &= 3. \end{align*}
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