Indices and surds

The number 32 can be factored into 16 * 2. Thus, sqrt(32) = sqrt(16) * sqrt(2) = 4 * sqrt(2).

Expand the square using (a - b)^2 = a^2 - 2ab + b^2. Here, (2sqrt(3))^2 = 43 = 12, and 2 * (2sqrt(3)) * 6 = 24sqrt(3), and 6^2 = 36. So, 12 - 24sqrt(3) + 36 = 48 - 24sqrt(3).

When adding radicals with the same radicand, add the coefficients: 4sqrt(2) + 6sqrt(2) = (4 + 6)sqrt(2) = 10sqrt(2).

Simplify the right side: (16 / sqrt(8))^6 = (16 / (2sqrt(2)))^6 = (8 / sqrt(2))^6 = (4sqrt(2))^6. This simplifies to (2^2 * 2^0.5)^6 = (2^2.5)^6 = 2^15. Setting 2^(2x-1) = 2^15, we get 2x-1 = 15, so 2x = 16, x = 8.

3^(x+1) = 9 / sqrt(3) = 3^2 / 3^0.5 = 3^(2 - 0.5) = 3^1.5. Thus, x + 1 = 1.5, which means x = 0.5.

Since 81 = 3^4, the equation 3^x = 81 becomes 3^x = 3^4. Therefore, x = 4.

4^x = 8 / sqrt(2) = 2^3 / 2^0.5 = 2^2.5. Since 4^x = (2^2)^x = 2^(2x), we have 2x = 2.5, so x = 1.25.

Since the bases are the same, equate the exponents: 2x = 8. Dividing by 2 gives x = 4.

Since the bases are the same, equate the exponents: x = 3.

(3/4)^2 = 3^2 / 4^2 = 9 / 16.

The cube root of x^2 is written as x^(2/3) using fractional exponents.

(3^2)^3 = 3^(2*3) = 3^6. Then 3^6 / 3^5 = 3^(6-5) = 3^1 = 3.

The square root of x is equivalent to x raised to the power of 1/2.

A negative exponent indicates a reciprocal: 6^(-3) = 1 / 6^3 = 1 / 216.

Any non-zero number raised to the power of 0 is 1.

Using the rule 1 / a^n = a^(-n), 1 / x^2 is equal to x^(-2).

The quotient rule for exponents states that a^m / a^n = a^(m-n). Here, a^6 / a^4 = a^(6-4) = a^2.

The power of a power rule states that (a^x)^y = a^(x*y). Multiplying the exponents gives a^(xy).

The product rule for exponents states that a^x * a^y = a^(x+y). Adding the exponents is the correct operation.