Answer:
25 x^2
Step-by-step explanation:
Simplify the following:
((5^(-2)/2^3)^4 2^28 x x)/((2^8/5^5×19^0)^2)
2^28 = (2^14)^2 = ((2^7)^2)^2 = ((2×2^6)^2)^2 = ((2 (2^3)^2)^2)^2 = ((2 (2×2^2)^2)^2)^2:
((5^(-2)/2^3)^4 ((2 (2×2^2)^2)^2)^2 x x)/(((2^8×19^0)/5^5)^2)
2^2 = 4:
((5^(-2)/2^3)^4 ((2 (2×4)^2)^2)^2 x x)/(((2^8×19^0)/5^5)^2)
2×4 = 8:
((5^(-2)/2^3)^4 ((2×8^2)^2)^2 x x)/(((2^8×19^0)/5^5)^2)
8^2 = 64:
((5^(-2)/2^3)^4 ((2×64)^2)^2 x x)/(((2^8×19^0)/5^5)^2)
2×64 = 128:
((5^(-2)/2^3)^4 (128^2)^2 x x)/(((2^8×19^0)/5^5)^2)
| | 1 | 2 | 8
× | | 1 | 2 | 8
| 1 | 0 | 2 | 4
| 2 | 5 | 6 | 0
1 | 2 | 8 | 0 | 0
1 | 6 | 3 | 8 | 4:
((5^(-2)/2^3)^4 16384^2 x x)/(((2^8×19^0)/5^5)^2)
| | | | 1 | 6 | 3 | 8 | 4
× | | | | 1 | 6 | 3 | 8 | 4
| | | | 6 | 5 | 5 | 3 | 6
| | 1 | 3 | 1 | 0 | 7 | 2 | 0
| | 4 | 9 | 1 | 5 | 2 | 0 | 0
| 9 | 8 | 3 | 0 | 4 | 0 | 0 | 0
1 | 6 | 3 | 8 | 4 | 0 | 0 | 0 | 0
2 | 6 | 8 | 4 | 3 | 5 | 4 | 5 | 6:
((5^(-2)/2^3)^4 268435456 x x)/(((2^8×19^0)/5^5)^2)
19^0 = 1:
((5^(-2)/2^3)^4 268435456 x x)/((2^8/5^5)^2)
5^5 = 5×5^4 = 5 (5^2)^2:
((5^(-2)/2^3)^4 268435456 x x)/((2^8/(5 (5^2)^2))^2)
5^2 = 25:
((5^(-2)/2^3)^4 268435456 x x)/((2^8/(5×25^2))^2)
| 2 | 5
× | 2 | 5
1 | 2 | 5
5 | 0 | 0
6 | 2 | 5:
((5^(-2)/2^3)^4 268435456 x x)/((2^8/(5×625))^2)
5×625 = 3125:
((5^(-2)/2^3)^4 268435456 x x)/((2^8/3125)^2)
2^8 = (2^4)^2 = ((2^2)^2)^2:
((5^(-2)/2^3)^4 268435456 x x)/((((2^2)^2)^2/3125)^2)
2^2 = 4:
((5^(-2)/2^3)^4 268435456 x x)/(((4^2)^2/3125)^2)
4^2 = 16:
((5^(-2)/2^3)^4 268435456 x x)/((16^2/3125)^2)
| 1 | 6
× | 1 | 6
| 9 | 6
1 | 6 | 0
2 | 5 | 6:
((5^(-2)/2^3)^4 268435456 x x)/((256/3125)^2)
(256/3125)^2 = 256^2/3125^2:
((5^(-2)/2^3)^4 268435456 x x)/(256^2/3125^2)
| | 2 | 5 | 6
× | | 2 | 5 | 6
| 1 | 5 | 3 | 6
1 | 2 | 8 | 0 | 0
5 | 1 | 2 | 0 | 0
6 | 5 | 5 | 3 | 6:
((5^(-2)/2^3)^4 268435456 x x)/(65536/3125^2)
| | | 3 | 1 | 2 | 5
× | | | 3 | 1 | 2 | 5
| | 1 | 5 | 6 | 2 | 5
| | 6 | 2 | 5 | 0 | 0
| 3 | 1 | 2 | 5 | 0 | 0
9 | 3 | 7 | 5 | 0 | 0 | 0
9 | 7 | 6 | 5 | 6 | 2 | 5:
((5^(-2)/2^3)^4 268435456 x x)/(65536/9765625)
5^(-2) = 1/25:
(((1/25)/2^3)^4 268435456 x x)/(65536/9765625)
2^3 = 2×2^2:
(((1/25)/(2×2^2))^4 268435456 x x)/(65536/9765625)
2^2 = 4:
(((1/25)/(2×4))^4 268435456 x x)/(65536/9765625)
2×4 = 8:
(((1/25)/8)^4 268435456 x x)/(65536/9765625)
Multiply exponents. (1/(25×8))^4 = (25×8)^(-4):
((25×8)^(-4) 268435456 x x)/(65536/9765625)
Multiply each exponent in 25×8 by -4:
(268435456 x x)/(65536/9765625) 1/25^4×1/8^4
8^4 = (8^2)^2:
((268435456 x x)/(25^4 (8^2)^2))/(65536/9765625)
8^2 = 64:
((268435456 x x)/(25^4×64^2))/(65536/9765625)
| | 6 | 4
× | | 6 | 4
| 2 | 5 | 6
3 | 8 | 4 | 0
4 | 0 | 9 | 6:
((268435456 x x)/(25^4×4096))/(65536/9765625)
25^4 = (25^2)^2:
((268435456 x x)/((25^2)^2 4096))/(65536/9765625)
| 2 | 5
× | 2 | 5
1 | 2 | 5
5 | 0 | 0
6 | 2 | 5:
((268435456 x x)/(625^2×4096))/(65536/9765625)
| | | 6 | 2 | 5
× | | | 6 | 2 | 5
| | 3 | 1 | 2 | 5
| 1 | 2 | 5 | 0 | 0
3 | 7 | 5 | 0 | 0 | 0
3 | 9 | 0 | 6 | 2 | 5:
((268435456 x x)/(390625×4096))/(65536/9765625)
390625×4096 = 1600000000:
((268435456 x x)/1600000000)/(65536/9765625)
The gcd of 268435456 and 1600000000 is 4096, so (x x×268435456)/1600000000 = ((4096×65536) x x)/(4096×390625) = 4096/4096×(65536 x x)/390625 = (65536 x x)/390625:
((65536 x x)/390625)/(65536/9765625)
Combine powers. (65536 x x)/390625 = (65536 x^(1 + 1))/390625:
((65536 x^(1 + 1))/390625)/(65536/9765625)
Multiply the numerator by the reciprocal of the denominator, ((65536 x^(1 + 1))/390625)/(65536/9765625) = (65536 x^(1 + 1))/390625×9765625/65536:
(65536×9765625 x^(1 + 1))/(390625×65536)
1 + 1 = 2:
(65536×9765625 x^2)/(390625×65536)
(65536 x^2×9765625)/(390625×65536) = 65536/65536×(x^2×9765625)/390625 = (x^2×9765625)/390625:
(9765625 x^2)/390625
| | | | |
3 | 9 | 0 | 6 | 2 | 5 | | | | | | 2 | 5
| 9 | 7 | 6 | 5 | 6 | 2 | 5
- | 7 | 8 | 1 | 2 | 5 | 0 |
| 1 | 9 | 5 | 3 | 1 | 2 | 5
- | 1 | 9 | 5 | 3 | 1 | 2 | 5
| | | | | | | 0:
Answer: 25 x^2
