Question

4^8*12^7*9^3/6^12*16^4=? 21^8*27^5*49^6/9^11*343^7=? 25^11*81^4/625^4*15^5*9^6=? 32^9*125^8/8^13*10^7*25^8=?

Answer 1

Решение во вложении:

Answer 2

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

1. 4^8 * 12^7 * 9^3 / 6^12 * 16^4

First, let's rewrite the numbers in terms of their prime factors: 4 = 2^2 12 = 2^2 * 3 9 = 3^2 6 = 2 * 3 16 = 2^4

Now, the expression becomes: (2^2)^8 * (2^2 * 3)^7 * (3^2)^3 / (2 * 3)^12 * 2^4^4

Now, apply the properties of exponents (a^m * a^n = a^(m+n) and (a^m)^n = a^(m*n)):

2^(28) * 2^(27) * 3^(7) * 3^(23) / 2^(112) * 3^(112) * 2^(44)

Simplify further:

2^16 * 2^14 * 3^7 * 3^6 / 2^12 * 3^12 * 2^16

Now, apply the property (a^m / a^n = a^(m-n)):

2^(16+14-12) * 3^(7+6-12)

Finally:

2^18 * 3^1

And the answer is:

2^18 * 3 = 262144 * 3 = 786432

2. 21^8 * 27^5 * 49^6 / 9^11 * 343^7

Let's rewrite the numbers in terms of their prime factors: 21 = 3 * 7 27 = 3^3 49 = 7^2 9 = 3^2 343 = 7^3

Now, the expression becomes: (37)^8 * 3^5 * (7^2)^6 / 3^(211) * (7^3)^7

Apply the properties of exponents:

3^8 * 7^8 * 3^5 * 7^12 / 3^22 * 7^21

Now, apply the property (a^m / a^n = a^(m-n)):

3^(8+5-22) * 7^(8+12-21)

Finally:

3^-9 * 7^-1 = 1 / (3^9 * 7)

And the answer is:

1 / (19683 * 7) = 1 / 137781

3. 25^11 * 81^4 / 625^4 * 15^5 * 9^6

Let's rewrite the numbers in terms of their prime factors: 25 = 5^2 81 = 3^4 625 = 5^4 15 = 3 * 5 9 = 3^2

Now, the expression becomes: (5^2)^11 * (3^4)^4 / (5^4)^4 * (3*5)^5 * (3^2)^6

Apply the properties of exponents:

5^(211) * 3^(44) / 5^(4*4) * 3^(5+6)

Now, apply the property (a^m / a^n = a^(m-n)):

5^(22-16) * 3^(16+11)

Finally:

5^6 * 3^27

And the answer is:

5^6 * 3^27 ≈ 16990566081 * 762939453125 ≈ 129600000000000000000

4. 32^9 * 125^8 / 8^13 * 10^7 * 25^8

Let's rewrite the numbers in terms of their prime factors: 32 = 2^5 125 = 5^3 8 = 2^3 10 = 2 * 5 25 = 5^2

Now, the expression becomes: (2^5)^9 * (5^3)^8 / (2^3)^13 * (2*5)^7 * (5^2)^8

Apply the properties of exponents:

2^(59) * 5^(38) / 2^(313) * 2^7 * 5^(28)

Now, apply the property (a^m / a^n = a^(m-n)):

2^(45-39) * 5^(24-39)

Finally:

2^6 * 5^-15 = 64 / 15

And the answer is:

64 / 15 ≈ 4.2667