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\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 470 with 8. Write the result 3760 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\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 470 with 0. Write the result 0 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\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 470 with 7. Write the result 3290 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\phantom{99}\\\phantom{\times}2820\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 470 with 6. Write the result 2820 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\phantom{99}\\\phantom{\times}2820\phantom{999}\\\phantom{\times}1410\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 470 with 3. Write the result 1410 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\phantom{99}\\\phantom{\times}2820\phantom{999}\\\phantom{\times}1410\phantom{9999}\\\phantom{\times}1880\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 470 with 4. Write the result 1880 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\phantom{99}\\\phantom{\times}2820\phantom{999}\\\phantom{\times}1410\phantom{9999}\\\phantom{\times}1880\phantom{99999}\\\phantom{\times}2350\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 470 with 5. Write the result 2350 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\phantom{99}\\\phantom{\times}2820\phantom{999}\\\phantom{\times}1410\phantom{9999}\\\phantom{\times}1880\phantom{99999}\\\phantom{\times}2350\phantom{999999}\\\underline{\phantom{\times}1410\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 470 with 3. Write the result 1410 at the end leaving 7 spaces to the right like this.
\begin{array}{c}\phantom{\times}470\\\underline{\times\phantom{}35436708}\\\phantom{\times}3760\\\phantom{\times}0\phantom{9}\\\phantom{\times}3290\phantom{99}\\\phantom{\times}2820\phantom{999}\\\phantom{\times}1410\phantom{9999}\\\phantom{\times}1880\phantom{99999}\\\phantom{\times}2350\phantom{999999}\\\underline{\phantom{\times}1410\phantom{9999999}}\\\phantom{\times}-524616424\end{array}
Now add the intermediate results to get final answer.