In order to simplify radical expressions, you need to be aware of the following rules and properties of radicals 1) From definition of n th root(s) and principal root Examples More examples on Roots of Real Numbers and Radicals. 2) Product (Multiplication) formula of radicals with equal indices is given by More examples on how to Multiply Radical Expressions. 3) Quotient (Division) formula of radicals with equal indices is given by More examples on how to Divide Radical Expressions. 4) You may add or subtract like radicals only Example More examples on how to Add Radical Expressions. 5) You may rewrite expressions without radicals (to rationalize denominators) as follows A) Example 1: B) Example 2: C) Example 3: More examples on how to Rationalize Denominators of Radical Expressions. Examples Rationalize and simplify the given expressions Answers to the above examples 1) Write 128 and 32 as product/powers of prime factors: 128 = 27 , 32 = 25 hence 2) Use product rule to write that √2 √6 = √12 3) Write 14 and 63 as products of prime numbers 14 = 2 × 7 , 63 = 32 × 7 and substitute 4) Write 32 and 16 as products of prime numbers 32 = 2 5 , 16 = 2 4 and substitute 5) Write 64 as products of prime numbers 64 = 2 6 and substitute Rationalize the denominator by multiplying numerator and denominator by (3√7)2 6) Write 54 as products of prime numbers 54 = 2 × 3 3 and substitute 7) Multiply the denominator and numerator by the conjugate of the denominator Expand and simplify More Questions With Answers Use all the rules and properties of radicals to rationalize and simplify the following expressions. Solutions to the Above Questions Write 25 and 125 as the product of prime factors: 25 = 52 and 125 = 53, hence Write 64 and 16 as the product of prime factors: 64 = 26 and 16 = 24, hence Use product rule Convert the mixed number under the radical into a fraction and substitute Use the division formula for radicals Write 64 and 27 as product of prime factors, substitute and simplify Use the product formula and write 34 as the product of prime factors Simplify For √(17 x) and √(34 x) to be real numbers, x must be positive hence |x| = x Write the radicand as a square and simplify Write the radicand as the product of $2$ and a square and simplify Simplify the radicand Write as the product of prime factors and simplify Since n is a positive integer, then N = 2 n + 1 is an odd integer. Hence Since n is a positive integer, then N = 2 n is an even integer. Hence Use division rule and simplify the radicand Multiply numerator and denominator by the conjugate of the denominator Expand and simplify