Roots and surds

A surd is an irrational root of a rational number. The sum of two surds, such as (1 + sqrt(2)) + (1 - sqrt(2)), can result in a rational number, but it can also result in an irrational number, such as sqrt(2) + sqrt(3). Therefore, the sum is not always irrational, never irrational, or always a surd.

The area is calculated by multiplying the base and height (or sides in this context), which is sqrt(6) * (6 - sqrt(6)). Distributing the sqrt(6) gives sqrt(6)*6 - sqrt(6)sqrt(6), which simplifies to 6sqrt(6) - 6.

Given a = sqrt(3) and b = cbrt(4). The expression is a^2 + b^3 * (1/a)^-2. This becomes (sqrt(3))^2 + (cbrt(4))^3 * (a^2). Substituting values: 3 + 4 * (sqrt(3))^2 = 3 + 4 * 3 = 3 + 12 = 15.

Simplify the radicals: sqrt(24) = sqrt(4 * 6) = 2sqrt(6) and sqrt(54) = sqrt(9 * 6) = 3sqrt(6). Adding them together: 2sqrt(6) + 3sqrt(6) = 5*sqrt(6).

Convert all to square roots: sqrt(6) is sqrt(6), 2sqrt(3) = sqrt(4 * 3) = sqrt(12), sqrt(15) is sqrt(15), and 3sqrt(2) = sqrt(9 * 2) = sqrt(18). Since 18 is the largest, 3*sqrt(2) is the biggest.

The volume of a cube is s^3 = 27. Thus, s = cbrt(27) = 3 meters. Converting 3 meters to centimeters gives 3 * 100 = 300 cm.

For a square with area 100km², each side is √100 = 10km = 10,000m. Option B correctly states this as 10^4 m (which equals 10,000m).

sqrt(2) is irrational. sqrt(4) = 2 (rational). sqrt(2 1/4) = sqrt(9/4) = 3/2 = 1.5 (rational). cbrt(27) = 3 (rational). Only one number is irrational.

sqrt(18) is irrational. cbrt(-8) = -2 (integer). sqrt(8) is irrational. cbrt(-81) is irrational. sqrt(81) = 9 (integer). There are two integers: -2 and 9.

The equation x^2 = 196 has two solutions: x = 14 and x = -14.

√81 + √64 = 9 + 8 = 17. We cannot combine the radicals as √(81+64). Option C correctly shows this as the sum 9+8.

sqrt(5 4/9) = sqrt(49/9). The square root of 49/9 is 7/3, which equals 2 1/3.