Evaluate
123456665543211
Factor
3^{5}\times 7\times 11\times 13\times 37\times 3607\times 3803
Share
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\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 9. Write the result 8999991 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\end{array}
Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 8. Write the result 7999992 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\end{array}
Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 7. Write the result 6999993 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\end{array}
Now multiply the first number with the 4^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 6. Write the result 5999994 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\end{array}
Now multiply the first number with the 5^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 5. Write the result 4999995 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\end{array}
Now multiply the first number with the 6^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 4. Write the result 3999996 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\end{array}
Now multiply the first number with the 7^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 3. Write the result 2999997 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\phantom{\times}1999998\phantom{9999999}\\\end{array}
Now multiply the first number with the 8^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 2. Write the result 1999998 at the end leaving 7 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\phantom{\times}1999998\phantom{9999999}\\\underline{\phantom{\times}999999\phantom{99999999}}\\\end{array}
Now multiply the first number with the 9^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 1. Write the result 999999 at the end leaving 8 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\phantom{\times}1999998\phantom{9999999}\\\underline{\phantom{\times}999999\phantom{99999999}}\\\phantom{\times}2125586987\end{array}
Now add the intermediate results to get final answer.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}