Verified answer

Note that

(a + b)⁴ = a⁴ + 4 a³ b + 6 a² b² + 4 a b³ + b⁴, and

(a - b)⁴ = a⁴ - 4 a³ b + 6 a² b² - 4 a b³ + b⁴.

So

(a + b)⁴ + (a - b)⁴ = 2 ( a⁴ + 6 a² b² + b⁴ ).

Here a = 5, b = 2√3. So

(5 + 2√3)⁴ + (5 - 2√3)⁴ = 2 ( 5⁴ + 6·5²·(2√3)² + (2√3)⁴ ) = 2 ( 5⁴ + 6·5²·12 + (12)² ),

which is clearly an integer.

Okay, so remember that 5 to the power of 5 to the power of 5, is completely the same as:

5 to the power of the product of the two powers. In this case, it will be 5 to the power of 25.

Same goes for the 5 to the power of 4 to the power of 6. It is 5 to the 24.

Multiply by 2, and we get 2(5^24).

So we have 5^25 - 2(5^24), and all of this divided by 5 to the 23.

Let us punch that into a calculator then... and it gives us 15!

The techniques I used can be found online, they are called Indices Laws.

The number or power at the top of the number can be called an index or exponent.

The number being raised to the index / exponent is called the base.

If you take a base-exponent to another exponent, then you can simply multiply the exponents together and keep the same base.

(a^x)^y = a^(xy)

And if you divide two numbers with the same base, then you can subtract the exponents:

a^x / a^y = a^(x - y)

So here:

[(5^5)^5 - 2(5^4)^6] / 5²³

You are doing that twice with your base 5s. multiply those exponents now and we get:

[5^25 - 2(5^24)] / 5²³

Let's factor out a 5^24 from the numerator, then we have:

(5^24)(5 - 2) / 5²³

Let's simplify the non-exponent term now:

(5^24)(5 - 2) / 5²³

(5^24)(3) / 5²³

Now, we have three terms either being multiplied or divided, so they can be done in any order and it won't change the solution. Let's divide our base 5-exponents by subtracting them.

5^(24 - 23) * 3

5 * 3

15

[(5^5)^5 - 2(5^4)^6]/5^23

= [5^(25) - 2*5^(24)]/5^23

= 3*5^(24)/5^(23) = 3*5 = 15

[(5^5)^5 - 2(5^4)^6] ÷ 5^23

= [(5^25) - 2(5^24)] ÷ 5^23

= (5^24)[(5) - 2(1)] ÷ 5^23

= (5)[(3)]

= 15

[(5⁵)⁵ − 2(5⁴)⁶] ÷ 5²³

[(5^25 - 2(5^24)] / 5^23

Factorise

5^24(5^1 - 2] /5^23

5^1[5^1 - 2) / 1

5(5 - 2)

5(3) = 15!!!!

= 15

Edit: [(5⁵)⁵ − 2(5⁴)⁶] ÷ 5²³

= 5^(-23) * (5^25 - 2*5^24)

= 5^2 - 2*5 = 15

Remember that (5^a)^b = 5^(ab). So:

[(5⁵)⁵ − 2(5⁴)⁶] ÷ 5²³ = [5^2²⁵ - 2 * 5^2²⁴] ÷ 5²³

= 5^2²⁴[5 - 2] ÷ 5²³ = 5 (3) = 15

Dear Darkx,

((5^(5))^(5)-2(5^(4))^(6))/(5^(23))

Raising a number to the 5th power is the same as multiplying the number by itself 5 times. In this case, 5 raised to the 5th power is 3125.

((3125)^(5)-2(5^(4))^(6))/(5^(23))

Raising a number to the 5th power is the same as multiplying the number by itself 5 times. In this case, 3125 raised to the 5th power is 2.9802*10^(17).

((2.9802*10^(17))-2(5^(4))^(6))/(5^(23))

Raising a number to the 4th power is the same as multiplying the number by itself 4 times. In this case, 5 raised to the 4th power is 625.

((2.9802*10^(17))-2(625)^(6))/(5^(23))

Raising a number to the 6th power is the same as multiplying the number by itself 6 times. In this case, 625 raised to the 6th power is 5.9605*10^(16).

((2.9802*10^(17))+(-2*5.9605*10^(16)))/(5^(23))

Multiply -2 by 5.9605*10^(16) to get -1.1921*10^(17).

((2.9802*10^(17))+(-1.1921*10^(17)))/(5^(23))

Remove the parentheses that are not needed from the expression.

(2.9802*10^(17)-1.1921*10^(17))/(5^(23))

Subtract 1.1921*10^(17) from 2.9802*10^(17) to get 1.7881*10^(17).

(1.7881*10^(17))/(5^(23))

Reduce the expression (1.7881*10^(17))/(5^(23)) by removing a factor of from the numerator and denominator.

15

=========================

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We know that (5^a)^b = 5^(ab). So:

[(5⁵)⁵ − 2(5⁴)⁶] ÷ 5²³ = [5^2²⁵ - 2 * 5^2²⁴] ÷ 5²³

= 5^2²⁴[5 - 2] ÷ 5²³

= 5 (3)

= 15