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
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
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.
Rationalize and simplify the given expressions
Answers to the above examples
Write 128 and 32 as product/powers of prime factors: 128 = 27 , 32 = 25 hence
Use product rule to write that √2 √6 = √12
Write 14 and 63 as products of prime numbers 14 = 2 × 7 , 63 = 32 × 7 and substitute
Write 32 and 16 as products of prime numbers 32 = 2 5 , 16 = 2 4 and substitute
Write 64 as products of prime numbers 64 = 2 6 and substitute
Rationalize the denominator by multiplying numerator and denominator by (3√7)2
Write 54 as products of prime numbers 54 = 2 × 3 3 and substitute
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
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