Fractional Indices and Surds

The square root of a negative number is not a real number, as no real number squared results in a negative value.

Simplify sqrt(18) = 3sqrt(2). Then 3sqrt(2) + sqrt(2) = 4*sqrt(2).

sqrt(18) = sqrt(9 * 2) = sqrt(9) * sqrt(2) = 3*sqrt(2).

By definition, the square of a square root of a non-negative number x is x itself: (sqrt(x))^2 = x.

sqrt(30) / sqrt(5) = sqrt(30 / 5) = sqrt(6).

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

Combine like terms: (3sqrt(7) + 2sqrt(7)) - 2sqrt(6) = 5sqrt(7) - 2*sqrt(6).

To simplify expressions with like radicals, subtract the coefficients while keeping the radical part the same. Here, 11 - 4 = 7, so the result is 7√3.

Like radicals are added by summing their coefficients. 2 + 3 = 5, resulting in 5√5.

The cube root of 27 is the number that, when multiplied by itself three times, equals 27. Since 3 * 3 * 3 = 27, the answer is 3.

The square root of 144 is the number that, when multiplied by itself, equals 144. Since 12 * 12 = 144, the answer is 12.

Using the power of a power rule, (a^m)^n = a^(m * n). Here, y^(2 * 1/4) = y^(2/4) = y^(1/2).

Using the quotient rule for exponents, x^a / x^b = x^(a - b). Here, 1/2 - 1/3 = 3/6 - 2/6 = 1/6, so the result is x^(1/6).

Using the product rule for exponents, x^a * x^b = x^(a + b). Here, 1/2 + 3/2 = 4/2 = 2, so the result is a^2.

The expression 16^(1/4) is equivalent to the fourth root of 16. Since 2 * 2 * 2 * 2 = 16, the answer is 2.

The fractional exponent 1/n corresponds to the nth root. Therefore, y^(1/4) is the fourth root of y, written as ∜y.

The fractional exponent 1/2 corresponds to the square root. Therefore, x^(1/2) is the square root of x, written as √x.