Index Law and Surds

(3*sqrt(5))^2 = 3^2 * (sqrt(5))^2 = 9 * 5 = 45.

(sqrt(6))^2 = 6 by definition of the square root.

6sqrt(12) * 2sqrt(6) = 12 * sqrt(72) = 12 * sqrt(36 * 2) = 12 * 6 * sqrt(2) = 72*sqrt(2).

5sqrt(3) * 8sqrt(5) = (5 * 8) * sqrt(3 * 5) = 40*sqrt(15).

sqrt(11) * sqrt(7) = sqrt(11 * 7) = sqrt(77).

5 * sqrt(x^3 * y^5) = 5 * sqrt(x^2 * x * y^4 * y) = 5 * x * y^2 * sqrt(x * y).

sqrt(384) = sqrt(64 * 6) = 8*sqrt(6).

4 * sqrt(100) = 4 * 10 = 40.

sqrt(6) + sqrt(6) = 2*sqrt(6).

The square root of 16 is 4.

(1/5)^-1 = (5/1)^1 = 5.

This is a difference of squares: (a + b)(a - b) = a^2 - b^2. Here, a = r^3 and b = s^3. So, (r^3)^2 - (s^3)^2 = r^6 - s^6.

(m^3 * n^-2)^-7 / (m^-5 * n^3)^4 = (m^-21 * n^14) / (m^-20 * n^12) = m^(-21 - (-20)) * n^(14 - 12) = m^-1 * n^2 = n^2 / m.

Divide the coefficients (2/6 = 1/3) and subtract the exponents for like bases (m^(2-5) = m^-3 and n^(-4 - -1) = n^-3). Since the indices must be positive, move the negative powers to the denominator to get 1/(3m^3n^3).

Let u = 2^x. The equation becomes u^2 - 5u + 4 = 0, which factors to (u-4)(u-1) = 0. Solving for u gives 2^x = 4 (so x = 2) or 2^x = 1 (so x = 0).

Any non-zero expression raised to the power of 0 is 1. Therefore, 4 * (anything)^0 = 4 * 1 = 4.

Divide the coefficients (14/28 = 1/2) and subtract the exponents for each variable: m^(3-2) = m, n^(4-2) = n^2, and p^(2-2) = p^0 = 1. This results in (mn^2)/2.

Divide the coefficients (7/1 = 7), subtract exponents for b (b^(5-2) = b^3), and note that a/a = 1 and c^4/c^4 = 1. The result is 7b^3.

Multiply the coefficients (2 * 4 * 1/2 = 4) and add the exponents for like bases (x^(3+1+4) = x^8 and y^(2+4) = y^6). The result is 4x^8y^6.

When multiplying terms with the same base, add the exponents: a^3 * a^4 = a^(3+4) = a^7.