You can always rewrite or change a radical expression into an exponent. For example, the expression ^(3)√27 (cube root of 27) can be rewritten into 27^(1/3). The reciprocal of the index number becomes the exponent to the radicand. The square root of √x is equivalent to x^1/2. Therefore, if you multiply √x^(2), you will have x^ (1/2)(2), which equals x.
Distributing radical equations is similar to standard distribution. For example, the expression ^4√xy is equal to ^4√x * ^4√y. Similarly, the same concept applies to radical fractions., e.g., ^3√(x/y) is equivalent to (^3√x)/( 3√y). When distributing ^3√(x^4), multiply the reciprocal of the index number by the radicand's exponent to make x^(4/3). Additionally, when you multiply a radical expression by itself, the radical cancels out. For example, √6 * √6= √36 or 6.
If you see a radical in the denominator of a fraction, you must eliminate it. Practice rationalizing by solving: 4/√a. Multiply both the numerator and denominator by the radical expression: 4*√a/(√a*√a), which is equal to 4√a/a. For a more complex expression, such as 3/ (√2+1), you must multiply both the numerator and denominator by (√2 -1), to cancel out the radicals. So, 3 *(√2 -1)/ (√2 +1)(√2 -1) = (3√2 -3)/ (2+√2-√2-1) or (3√2 -3).
You can not have negative radicands, unless the radical has an odd index number. For example, √-4 has no solution because the square of two numbers can never be negative. However, ^3√-27 has the solution of -3, because -3 *-3 *-3 is equal to -27. The positive root of a number is written as the following, √36, or 6, whereas the negative root of a number is written as -√36, or -6.