$3$ . Find the square root of each of the following by division. $\newline$ (i) $841$ $\newline$ (ii) $2304$ $\newline$ (iii) $39204$ $\newline$ (iv) $55225$ $\newline$ (v) $177241$ $\newline$ (vi) $425104$

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Algebra 2

Multiplication with rational exponents

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3

. Find the square root of each of the following by division.

\newline

(i)

841

\newline

(ii)

2304

\newline

(iii)

39204

\newline

(iv)

55225

\newline

(v)

177241

\newline

(vi)

425104

Group Digits into Pairs: Group the digits of $841$ into pairs from right to left, so we have $08$ and $41$ . Find the largest square number less than or equal to $8$ , which is $4$ ( $2^2$ ). Place $2$ as the first digit of the square root and subtract $4$ from $8$ , leaving $4$ . Bring down the next pair of digits, $41$ , to make $08$ $1$ . Double the current quotient ( $2$ ), which gives us $4$ , and find a digit ( $08$ $4$ ) such that $08$ $5$ is less than or equal to $08$ $1$ . $08$ $7$ works because $08$ $8$ . Place $08$ $7$ next to $2$ to get $41$ $1$ as the quotient. Subtract $08$ $1$ from $08$ $1$ to get $41$ $4$ .
Find Largest Square Number: Group the digits of $2304$ into pairs, so we have $23$ and $04$ . Find the largest square number less than or equal to $23$ , which is $16$ ( $4^2$ ). Place $4$ as the first digit of the square root and subtract $16$ from $23$ , leaving $7$ . Bring down the next pair of digits, $04$ , to make $23$ $1$ . Double the current quotient ( $4$ ), which gives us $23$ $3$ , and find a digit ( $23$ $4$ ) such that $23$ $5$ is less than or equal to $23$ $1$ . $23$ $3$ works because $23$ $8$ . Place $23$ $3$ next to $4$ to get $04$ $1$ as the quotient. Subtract $23$ $1$ from $23$ $1$ to get $04$ $4$ .
Place First Digit: Group the digits of $39204$ into pairs, so we have $03$ , $92$ , and $04$ . Find the largest square number less than or equal to $3$ , which is $1$ ( $1^2$ ). Place $1$ as the first digit of the square root and subtract $1$ from $3$ , leaving $03$ $0$ . Bring down the next pair of digits, $92$ , to make $03$ $2$ . Double the current quotient ( $1$ ), which gives us $03$ $0$ , and find a digit ( $03$ $5$ ) such that $03$ $6$ is less than or equal to $03$ $2$ . $03$ $8$ works because $03$ $9$ . Place $03$ $8$ next to $1$ to get $92$ $2$ as the quotient. Subtract $92$ $3$ from $03$ $2$ to get $92$ $5$ . Bring down the next pair of digits, $04$ , to make $92$ $7$ . Double the current quotient ( $92$ $2$ ), which gives us $92$ $9$ , and find a digit ( $03$ $5$ ) such that $04$ $1$ is less than or equal to $92$ $7$ . $04$ $3$ works because $04$ $4$ . Place $04$ $3$ next to $92$ $2$ to get $04$ $7$ as the quotient. Subtract $92$ $7$ from $92$ $7$ to get $3$ $0$ .
Bring Down Next Pair: Group the digits of $55225$ into pairs, so we have $05$ , $52$ , and $25$ . Find the largest square number less than or equal to $5$ , which is $4$ ( $2^2$ ). Place $2$ as the first digit of the square root and subtract $4$ from $5$ , leaving $05$ $0$ . Bring down the next pair of digits, $52$ , to make $05$ $2$ . Double the current quotient ( $2$ ), which gives us $4$ , and find a digit ( $05$ $5$ ) such that $05$ $6$ is less than or equal to $05$ $2$ . $05$ $8$ works because $05$ $9$ . Place $05$ $8$ next to $2$ to get $52$ $2$ as the quotient. Subtract $52$ $3$ from $05$ $2$ to get $52$ $2$ . Bring down the next pair of digits, $25$ , to make $52$ $7$ . Double the current quotient ( $52$ $2$ ), which gives us $52$ $9$ , and find a digit ( $05$ $5$ ) such that $25$ $1$ is less than or equal to $52$ $7$ . $5$ works because $25$ $4$ . Place $5$ next to $52$ $2$ to get $25$ $7$ as the quotient. Subtract $52$ $7$ from $52$ $7$ to get $5$ $0$ .
Double Current Quotient: Group the digits of $177241$ into pairs, so we have $01$ , $77$ , and $24$ . Find the largest square number less than or equal to $1$ , which is $1$ ( $1^2$ ). Place $1$ as the first digit of the square root and subtract $1$ from $1$ , leaving $01$ $0$ . Bring down the next pair of digits, $77$ , to make $01$ $2$ . Double the current quotient ( $1$ ), which gives us $01$ $4$ , and find a digit ( $01$ $5$ ) such that $01$ $6$ is less than or equal to $01$ $2$ . $01$ $8$ works because $01$ $9$ . Place $01$ $8$ next to $1$ to get $77$ $2$ as the quotient. Subtract $77$ $3$ from $01$ $2$ to get $77$ $5$ . Bring down the next pair of digits, $24$ , to make $77$ $7$ . Double the current quotient ( $77$ $2$ ), which gives us $77$ $9$ , and find a digit ( $01$ $5$ ) such that $24$ $1$ is less than or equal to $77$ $7$ . $1$ works because $24$ $4$ . Place $1$ next to $77$ $2$ to get $24$ $7$ as the quotient. Subtract $24$ $8$ from $77$ $7$ to get $1$ $0$ .