Respuesta :
Answer:
- √15625 = 125
- √21025 = 145
- √60025 = 245
- √42025 = 205
Step-by-step explanation:
You want the square roots of the perfect squares 15625, 21025, 60025, 42025.
Ones digit
You can find the appropriate ones digit by considering the units digit of the squares of the digits 0–9. The ending digits of these squares are ...
0² ends in 0; (1 or 9)² ends in 1; (2 or 8)² ends in 4; (3 or 7)² ends in 9
(4 or 6)² ends in 6; 5² ends in 5.
All of these squares end in 5, so their roots will also end in 5.
Hundreds digit
When digits are paired from right to left, starting at the decimal point, the leftmost digit (or pair) will be equal or larger than the square of the leftmost digit of the root. We know that 1² = 1, 2² = 4, 3² = 9, so the hundreds digits of the roots of these numbers will be 1 or 2.
Tens digit
The middle digit of the root can be found a couple of ways. Perhaps easiest is to use your knowledge of squares of numbers to 20 or 30. The left two digit pairs will have a square root that (mostly) matches the left two digits of the root.
As a check, you can "cast out nines." The sum of digits of the square will be the square of the sum of digits of the root, as reduced to a single digit.
Another way guess the tens digit of the root is to find the difference from subtracting the square of the hundreds digit from the leftmost digit of the square. Prepend that number to the next digit of the square, and divide the result by double the root's hundreds digit. The quotient will be equal or slightly greater than the tens digit of the root.
15625
The leftmost "pair" of digits is 01, which has a square root of 1. Subtracting 1² from 01, we get 0. Prepending this to the next digit gives 05, and dividing that by 2·1 gives 2 (remainder 1). We already know the units digit of the root is 5, so our first guess at the square root is ...
√15625 = 125
Check: sum of number digits = 1+5+6+2+5 = 19 ⇒ 1+9 = 10 ⇒ 1 +0 = 1
Sum of root digits = 1 +2 +5 = 8, and 8² = 64 ⇒ 6+4 = 10 ⇒ 1 +0 = 1
This tells us we have the correct root
21025
The leftmost "pair" of digits is 02, which has a square root of 1. Subtracting 1² from 02 gives a remainder of 1, and prepending that to the next digit gives 11. Dividing 11 by 2·1 gives 5, suggesting our root is 155. Since we know that 15² = 225 > 210, we know that 5 is too large for the tens digit. The "casting out nines" check confirms that the proper root is ...
√21025 = 145
Check: (1+4+5)≡1, 1² = 1; (2+1+0+2+5)≡1, so 145 is consistent with this check.
60025
The leftmost "pair" of digits is 06, which has a root of 2. Subtracting 2² gives a remainder of 2, and dividing 20 by 2·2 gives 5, suggesting our root is 255. Again, we know that 25² = 625, so 5 is too large. The proper root is ...
√60025 = 245
Check: (2+4+5)≡2, 2² = 4; (6+0+0+2+5)≡4, so 245 is consistent with this check.
42025
The first root digit will be 2. We next divide 02 by 4 to get 0, suggesting our root is ...
√42025 = 205
Check: (2+0+5)²≡4; (4+2+0+2+5)≡4, so this root value checks OK.
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Additional comment
The quickest check is to use a calculator to find the root. You can also square the number you have found as the root, and compare to the original. There are mental math procedures for computing the square digit by digit that can be relatively painless.
