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\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 5555599 with 8. Write the result 44444792 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\phantom{\times9}44444792\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 5555599 with 8. Write the result 44444792 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\phantom{\times9}44444792\phantom{9}\\\phantom{\times}27777995\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 5555599 with 5. Write the result 27777995 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\phantom{\times9}44444792\phantom{9}\\\phantom{\times}27777995\phantom{99}\\\phantom{\times}44444792\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 5555599 with 8. Write the result 44444792 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\phantom{\times9}44444792\phantom{9}\\\phantom{\times}27777995\phantom{99}\\\phantom{\times}44444792\phantom{999}\\\phantom{\times}44444792\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 5555599 with 8. Write the result 44444792 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\phantom{\times9}44444792\phantom{9}\\\phantom{\times}27777995\phantom{99}\\\phantom{\times}44444792\phantom{999}\\\phantom{\times}44444792\phantom{9999}\\\underline{\phantom{\times}27777995\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 5555599 with 5. Write the result 27777995 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5555599\\\underline{\times\phantom{9999}588588}\\\phantom{\times99}44444792\\\phantom{\times9}44444792\phantom{9}\\\phantom{\times}27777995\phantom{99}\\\phantom{\times}44444792\phantom{999}\\\phantom{\times}44444792\phantom{9999}\\\underline{\phantom{\times}27777995\phantom{99999}}\\\phantom{\times}1488791956\end{array}
Now add the intermediate results to get final answer.