Question

(2a3)4 · (3a2)3 = a (16a5)2 : (8a3)3 = a (6a4)2 : (2a2)3 = a (10a4)4 : (5a3)4 = a (4a3)3 : (5a4)2 = a (6a6)4 : (12a10)2 = a (12a3)2 : (4a2)2 = a (2a7)4 · (5a2)2 = a (5a3)2 · (2a4)3 = a (3a5)3 · (9a7)2 = a

Answer 1

Відповідь:

1.   $(2a^{3} )^{4}*(3a^{2})^{3}=2^{4}a^{3*4}*3^{3}a^{2*3}=16a^{12}*27a^{6}=16*27*a^{12+6}=432a^{18}$

Answer 2

Let's simplify each of the expressions step by step:

1. (2a^3)^4 · (3a^2)^3 = (2^4 * a^(34)) · (3^3 * a^(23)) = 16a^12 · 27a^6 = (16 * 27) * a^(12 + 6) = 432a^18

2. a(16a^5)^2 : (8a^3)^3 = a(16^2 * a^(52)) : (8^3 * a^(33)) = a(256a^10) : (512a^9) = (a * 256a^10) / 512a^9 = (256a^11) / 512a^9 = (256/512) * a^(11-9) = 1/2 * a^2

3. a(6a^4)^2 : (2a^2)^3 = a(36a^8) : (8a^6) = (a * 36a^8) / 8a^6 = (36a^9) / (8a^6) = (36/8) * a^(9-6) = 9/2 * a^3

4. a(10a^4)^4 : (5a^3)^4 = a(10^4 * a^(44)) : (5^4 * a^(34)) = a(10,000a^16) : (625a^12) = (a * 10,000a^16) / 625a^12 = (10,000a^17) / 625a^12 = (10,000/625) * a^(17-12) = 16a^5

5. a(4a^3)^3 : (5a^4)^2 = a(64a^9) : (25a^8) = (a * 64a^9) / 25a^8 = (64a^10) / 25a^8 = (64/25) * a^(10-8) = 64/25 * a^2

6. a(6a^6)^4 : (12a^10)^2 = a(1,296a^24) : (144a^20) = (a * 1,296a^24) / 144a^20 = (1,296a^25) / 144a^20 = (1,296/144) * a^(25-20) = 9a^5

7. a(12a^3)^2 : (4a^2)^2 = a(144a^6) : (16a^4) = (a * 144a^6) / 16a^4 = (144a^7) / 16a^4 = (144/16) * a^(7-4) = 9a^3

8. a(2a^7)^4 · (5a^2)^2 = a(16a^28) · (25a^4) = 16a^(28+1) * 25a^4 = 400a^29

9. a(5a^3)^2 · (2a^4)^3 = a(25a^6) · (8a^12) = 200a^(6+12) = 200a^18

10. a(3a^5)^3 · (9a^7)^2 = a(27a^15) · (81a^14) = 2187a^(15+14) = 2187a^29

So, the final simplified expression is:

432a^18 : (1/2)a^2 : 9/2a^3 : 16a^5 : 64/25a^2 : 9a^5 : 9a^3 : 400a^29 : 200a^18 : 2187a^29

Now, to simplify this combined expression, we can combine like terms by adding their coefficients if they have the same exponent "a". Let's do that:

432a^18 : (1/2)a^2 : 9/2a^3 : 16a^5 : 64/25a^2 : 9a^5 : 9a^3 : 400a^29 : 200a^18 : 2187a^29

= (432 + 200)a^18 : (1/2 * 64/25)a^2 : 9/2a^3 : (16 + 9)a^5 : 9a^3 : 400a^29 : 2187a^29

= 632a^18 : (32/25)a^2 : 9/2a^3 : 25a^5 : 9a^3 : 2187a^29

Now, we have simplified the expression as much as possible.