Solve 5544x54545454.sqrt{=1===}

$$5544x54545454. \sqrt{ =1=== }$$

Evaluate

$$5544x_{54545454}$$

Steps For Long Multiplication

First line up the numbers vertically and match the places from the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\end{array}$$

Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 4. Write the result 22176 at the end leaving 0 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\end{array}$$

Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 5. Write the result 27720 at the end leaving 1 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\end{array}$$

Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 4. Write the result 22176 at the end leaving 2 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\end{array}$$

Now multiply the first number with the 4^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 5. Write the result 27720 at the end leaving 3 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\phantom{\times99}27720\phantom{999}\\\end{array}$$

Now multiply the first number with the 5^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 4. Write the result 22176 at the end leaving 4 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\phantom{\times99}27720\phantom{999}\\\phantom{\times9}22176\phantom{9999}\\\end{array}$$

Now multiply the first number with the 6^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 5. Write the result 27720 at the end leaving 5 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\phantom{\times99}27720\phantom{999}\\\phantom{\times9}22176\phantom{9999}\\\phantom{\times}27720\phantom{99999}\\\end{array}$$

Now multiply the first number with the 7^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 4. Write the result 22176 at the end leaving 6 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\phantom{\times99}27720\phantom{999}\\\phantom{\times9}22176\phantom{9999}\\\phantom{\times}27720\phantom{99999}\\\phantom{\times}22176\phantom{999999}\\\end{array}$$

Now multiply the first number with the 8^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 5544 with 5. Write the result 27720 at the end leaving 7 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\phantom{\times99}27720\phantom{999}\\\phantom{\times9}22176\phantom{9999}\\\phantom{\times}27720\phantom{99999}\\\phantom{\times}22176\phantom{999999}\\\underline{\phantom{\times}27720\phantom{9999999}}\\\end{array}$$

Now add the intermediate results to get final answer.

$$\begin{array}{c}\phantom{\times999999}5544\\\underline{\times\phantom{99}54545454}\\\phantom{\times99999}22176\\\phantom{\times9999}27720\phantom{9}\\\phantom{\times999}22176\phantom{99}\\\phantom{\times99}27720\phantom{999}\\\phantom{\times9}22176\phantom{9999}\\\phantom{\times}27720\phantom{99999}\\\phantom{\times}22176\phantom{999999}\\\underline{\phantom{\times}27720\phantom{9999999}}\\\phantom{\times}1752286256\end{array}$$

Differentiate w.r.t. x_54545454

$$5544$$

Steps Using Definition of a Derivative

The derivative of ax^{n} is nax^{n-1}.

$$5544x_{54545454}^{1-1}$$

Subtract 1 from 1.

$$5544x_{54545454}^{0}$$

For any term t except 0, t^{0}=1.

$$5544\times 1$$

For any term t, t\times 1=t and 1t=t.

$$5544$$