Find its square root and raise it in the second power.
64 = 82
81 = 92
841 = ?2 you need to work a little bit here
1. base would be in two digits (odd number with 3 digits, (3+1)/2)
2. the first digit would be 2 (the square of 2 is less than 8), so the base would be between 20 and 30.
3. the second digit would be 9 (the square of 9 ends in 1)
4. the answer is 29, 841 = 292
529 = ?2
1. base would be in two digits (odd number with 3 digits, (3+1)/2)
2. the first digit would be 2 (the square of 2 is less than 5), so the base would be between 20 and 30.
3. the second digit would be 3 or 7 (the square of 3 and 7 end in 9); is it 23 or 27?
4. 529 - 2232 = 529 - 409 = 120; it is 3, because 2(2*3)=12. So 529 = 232
2809 = ?2
1. base would be in two digits (even number with 4 digits, 4/2)
2. the first digit would be 5 (the square of 5 is less than 28,), so the base would be between 50 and 60.
3. the second digit would be 3 or 7 (the square of 3 and 7 end in 9); is it 53 or 57?
4. 2809 - 5232 = 2809 - 2509 = 300; it is 3, because 2(5*3)=30. So 2809 = 532
7056 = ?2
1. base would be in two digits (even number with 4 digits, 4/2)
2. the first digit would be 8 (the square of 8 is less than 70,), so the base would be between 80 and 90.
3. the second digit would be 4 or 6 (the square of 4 and 6 end in 6); is it 84 or 86?
4. 7056 - 8242 = 7056 - 6416 = 640; it is 4, because 2(8*4)=64. So 7056 = 842
7396 = ?2
1. base would be in two digits (even number with 4 digits, 4/2)
2. the first digit would be 8 (the square of 8 is less than 73), so the base would be between 80 and 90.
3. the second digit would be 4 or 6 (the square of 4 and 6 end in 6); is it 84 or 86?
4. 7396 - 8262 = 7396 - 6436 = 960; it is 6, because 2(8*6)=96. So 7396
(if you test 4 first, you'll see that it doesn't work, so it will be 6)
55696 = ?2
1. base would be in three digits (odd number with 5 digits, (5+1)/2)
2. the first digit would be 2 (the square of 2 is less than 5), so the base would be between 200 and 300.
3. the last digit would be 4 or 6 (the square of 4 and 6 end in 6); is it 2_4 or 2_6?
4. let's work a little bit here:
2_4; 22_42 = 40016
40016+ ... = 45696
00080 (101*2*4*_, we need to find the middle digit)
01600 (102*2*2*4)
04000 (103*2*2*_)
55696 - 45696 = 10000
40016
00080
01600
04000
10000
00080 doesn't change, this tell us the middle digit can be 1 or 6. But one is too small and 6 is too big (2*2*1= 4, 2*2*6=32). So the last digit is not 4, it is 6. Let's find the middle digit.
2_6; 22_62 = 40036
40036+ ... = 46556
00120 (101*2*6*_, we need to find the middle digit)
02400 (102*2*2*6)
04000 (103*2*2*_)
55696 - 46556 = 9140
40036
00120
02400
04000
09140
00120 becomes 00160, and tell us the middle digit can be 3 or 8, but 8 is too big (2*2*8=32), so it is 3
Thus, 55696 = 2362.
263169 = ?2
1. base would be in three digits (even number with 6 digits, 6/2)
2. the first digit would be 5 (the square of 5 is less than 26), so the base would be between 500 and 600.
3. the second digit would be 3 or 7 (the square of 3 and 7 end in 9); is it 5_3 or 5_7?
4. let's work a little bit here:
5_3; 52_32 = 250009
250009+ ... = 263069
000060 (101*2*3*_, we need to find the middle digit)
003000 (102*2*5*3)
010000 (103*2*5*_)
263169 - 263069 = 100
250009
000060
003000
010000
000100
00060 doesn't change so the only possibility is when the middle digit is 1.
Thus, 2631169 = 5132
30 is not a perfect square. Its square root is a fraction and the square root of a perfect square is always an integer.
500 is not a perfect square. Its square root is a fraction and the square root of a perfect square is always an integer.
8 is a perfect cube; it is not a perfect square. Its square root is a fraction and the square root of a perfect square is always an integer....you're thinking of 9...
1764 was a perfect square year.
No, 8x8 is 64 and that is a perfect square... 69 is not!
192 = 361
128 is neither a perfect square nor a perfect cube, it is 2⁷ (2 to the 7th power).
Note that t15 contains the perfect square factor of t14. We can rewrite the expression as: √(t * t14) = √(t * (t7)²) = t7√t
And exponent of -2 represents the square root
Since 88 factors to 23 x 11, it is not a perfect square, it is not a perfect cube, etc. Since the square root of 88 is approximately 9.38, you can say that 88 is close to 9.382.
8 to the power of 2, like 8^2
The relevant exponent is -2.
Cut the exponent in half.
Any even exponent of 3 is a square number.
The square of a number can be represented by writing the number with an exponent of 2, whereas the square root of the number is represented by writing the number with an exponent of 1/2.
A perfect square is called so because it is the product of an integer multiplied by itself, resembling a geometric square with equal sides. In mathematical terms, a perfect square can be expressed as ( n^2 ), where ( n ) is a whole number. The exponent of 2 indicates that the base number is used twice in multiplication, highlighting its relationship to the geometry of squares. Thus, perfect squares always yield whole numbers when taking the square root, reinforcing their definition.
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