find the square roots of $100$ and $169$ by the method of repeated subtraction

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Algebra 2

Evaluate rational exponents

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Q. find the square roots of

100

and

169

by the method of repeated subtraction

Understanding Repeated Subtraction: Understand the method of repeated subtraction for finding square roots. $\newline$ The method of repeated subtraction involves subtracting consecutive odd numbers from the number until we reach $0$ . The number of times we subtract is the square root.
Subtracting Odd Numbers: Start with the number $100$ and subtract the smallest odd number, which is $1$ . $100 - 1 = 99$
Counting Subtractions: Subtract the next odd number, which is $3$ , from the remaining number. $\newline$ $99 - 3 = 96$
Finding Square Root of $100$ : Continue subtracting the next consecutive odd number, which is $5$ . $96 - 5 = 91$
Subtracting Odd Numbers: Subtract the next odd number, which is $7$ . $\newline$ $91 - 7 = 84$
Counting Subtractions: Subtract the next odd number, which is $9$ . $84 - 9 = 75$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $\newline$ $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $\newline$ $64 - 13 = 51$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $\newline$ $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $\newline$ $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $\newline$ $51 - 15 = 36$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $\newline$ $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $\newline$ $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $\newline$ $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $\newline$ $36 - 17 = 19$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ .
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number.
$75 - 11 = 64$ $8$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number.
$75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ .
$13$ $0$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number.
$75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ .
$13$ $0$ Subtract the next odd number, which is $13$ $1$ .
$13$ $2$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $13$ $4$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $13$ $4$ Subtract the next odd number, which is $11$ . $13$ $6$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $\newline$ $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $\newline$ $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $\newline$ $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $\newline$ $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $\newline$ $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . $\newline$ We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $\newline$ $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $\newline$ $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $\newline$ $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $\newline$ $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $\newline$ $13$ $4$ Subtract the next odd number, which is $11$ . $\newline$ $13$ $6$ Subtract the next odd number, which is $13$ . $\newline$ $13$ $8$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number.
$75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ .
$13$ $0$ Subtract the next odd number, which is $13$ $1$ .
$13$ $2$ Subtract the next odd number, which is $13$ $3$ .
$13$ $4$ Subtract the next odd number, which is $11$ .
$13$ $6$ Subtract the next odd number, which is $13$ .
$13$ $8$ Subtract the next odd number, which is $15$ .
$64 - 13 = 51$ $0$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $13$ $4$ Subtract the next odd number, which is $11$ . $13$ $6$ Subtract the next odd number, which is $13$ . $13$ $8$ Subtract the next odd number, which is $15$ . $64 - 13 = 51$ $0$ Subtract the next odd number, which is $17$ . $64 - 13 = 51$ $2$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $13$ $4$ Subtract the next odd number, which is $11$ . $13$ $6$ Subtract the next odd number, which is $13$ . $13$ $8$ Subtract the next odd number, which is $15$ . $64 - 13 = 51$ $0$ Subtract the next odd number, which is $17$ . $64 - 13 = 51$ $2$ Subtract the next odd number, which is $19$ . $64 - 13 = 51$ $4$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number.
$75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ .
$13$ $0$ Subtract the next odd number, which is $13$ $1$ .
$13$ $2$ Subtract the next odd number, which is $13$ $3$ .
$13$ $4$ Subtract the next odd number, which is $11$ .
$13$ $6$ Subtract the next odd number, which is $13$ .
$13$ $8$ Subtract the next odd number, which is $15$ .
$64 - 13 = 51$ $0$ Subtract the next odd number, which is $17$ .
$64 - 13 = 51$ $2$ Subtract the next odd number, which is $19$ .
$64 - 13 = 51$ $4$ Subtract the next odd number, which is $64 - 13 = 51$ $5$ .
$64 - 13 = 51$ $6$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $\newline$ $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $\newline$ $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $\newline$ $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $\newline$ $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $\newline$ $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . $\newline$ We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $\newline$ $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $\newline$ $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $\newline$ $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $\newline$ $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $\newline$ $13$ $4$ Subtract the next odd number, which is $11$ . $\newline$ $13$ $6$ Subtract the next odd number, which is $13$ . $\newline$ $13$ $8$ Subtract the next odd number, which is $15$ . $\newline$ $64 - 13 = 51$ $0$ Subtract the next odd number, which is $17$ . $\newline$ $64 - 13 = 51$ $2$ Subtract the next odd number, which is $19$ . $\newline$ $64 - 13 = 51$ $4$ Subtract the next odd number, which is $64 - 13 = 51$ $5$ . $\newline$ $64 - 13 = 51$ $6$ Subtract the next odd number, which is $64 - 13 = 51$ $7$ . $\newline$ $64 - 13 = 51$ $8$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ . $75 - 11 = 64$ Subtract the next odd number, which is $13$ . $64 - 13 = 51$ Subtract the next odd number, which is $15$ . $51 - 15 = 36$ Subtract the next odd number, which is $17$ . $36 - 17 = 19$ Subtract the next odd number, which is $19$ . $19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ . We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ . $75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number. $75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ . $13$ $0$ Subtract the next odd number, which is $13$ $1$ . $13$ $2$ Subtract the next odd number, which is $13$ $3$ . $13$ $4$ Subtract the next odd number, which is $11$ . $13$ $6$ Subtract the next odd number, which is $13$ . $13$ $8$ Subtract the next odd number, which is $15$ . $64 - 13 = 51$ $0$ Subtract the next odd number, which is $17$ . $64 - 13 = 51$ $2$ Subtract the next odd number, which is $19$ . $64 - 13 = 51$ $4$ Subtract the next odd number, which is $64 - 13 = 51$ $5$ . $64 - 13 = 51$ $6$ Subtract the next odd number, which is $64 - 13 = 51$ $7$ . $64 - 13 = 51$ $8$ Subtract the next odd number, which is $64 - 13 = 51$ $9$ . $15$ $0$
Finding Square Root of $169$ : Subtract the next odd number, which is $11$ .
$75 - 11 = 64$ Subtract the next odd number, which is $13$ .
$64 - 13 = 51$ Subtract the next odd number, which is $15$ .
$51 - 15 = 36$ Subtract the next odd number, which is $17$ .
$36 - 17 = 19$ Subtract the next odd number, which is $19$ .
$19 - 19 = 0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $75 - 11 = 64$ $1$ times, so the square root of $75 - 11 = 64$ $2$ is $75 - 11 = 64$ $1$ . Now, start with the number $75 - 11 = 64$ $4$ and subtract the smallest odd number, which is $75 - 11 = 64$ $5$ .
$75 - 11 = 64$ $6$ Subtract the next odd number, which is $75 - 11 = 64$ $7$ , from the remaining number.
$75 - 11 = 64$ $8$ Continue subtracting the next consecutive odd number, which is $75 - 11 = 64$ $9$ .
$13$ $0$ Subtract the next odd number, which is $13$ $1$ .
$13$ $2$ Subtract the next odd number, which is $13$ $3$ .
$13$ $4$ Subtract the next odd number, which is $11$ .
$13$ $6$ Subtract the next odd number, which is $13$ .
$13$ $8$ Subtract the next odd number, which is $15$ .
$64 - 13 = 51$ $0$ Subtract the next odd number, which is $17$ .
$64 - 13 = 51$ $2$ Subtract the next odd number, which is $19$ .
$64 - 13 = 51$ $4$ Subtract the next odd number, which is $64 - 13 = 51$ $5$ .
$64 - 13 = 51$ $6$ Subtract the next odd number, which is $64 - 13 = 51$ $7$ .
$64 - 13 = 51$ $8$ Subtract the next odd number, which is $64 - 13 = 51$ $9$ .
$15$ $0$ Count the number of subtractions performed to reach $75 - 11 = 64$ $0$ .
We have subtracted $13$ times, so the square root of $75 - 11 = 64$ $4$ is $13$ .

find the square roots of $100$ and $169$ by the method of repeated subtraction

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find the square roots of 100100100 and 169169169 by the method of repeated subtraction

find the square roots of $100$ and $169$ by the method of repeated subtraction