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3 years ago

(2^85^-519^0)^-2x(5^-2/2^3)^4 x 2^28

Mathematics

1 answer:

Free_Kalibri [48]3 years ago

4 0

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

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Complete question :

The Venn diagram relating to the question can be found in the picture attached below :

Answer:

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(2^8*5^-5*19^0)^-2x(5^-2/2^3)^4 x 2^28

(2^85^-519^0)^-2x(5^-2/2^3)^4 x 2^28