Simplify Radical Expressions
Questions with Solutions for Grade 10

Grade 10 questions on how to simplify radicals expressions with solutions are presented.

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

equation 1

Examples

equation 3


More examples on
Roots of Real Numbers and Radicals.

2) Product (Multiplication) formula of radicals with equal indices is given by

equation 4

More examples on how to
Multiply Radical Expressions.


3) Quotient (Division) formula of radicals with equal indices is given by

equation 5

More examples on how to
Divide Radical Expressions.


4) You may add or subtract like radicals only

Example:
equation 6


More examples on how to
Add Radical Expressions.


5) You may rewrite expressions without radicals (to rationalize denominators) as follows

A) Example 1:
equation 7
B) Example 2:
equation 9
C) Example 3:
equation 9


More examples on how to
Rationalize Denominators of Radical Expressions.


Examples


Rationalize and simplify the given expressions

equation 10


Answers to the above examples


1)
Write 128 and 32 as product/powers of prime factors: 128=2
7 , 32=25 hence

equation 11


2)
Use product rule to write that √2 √6 = √12

equation 12


3)
Write 14 and 63 as products of prime numbers 14=2 * 7 , 63 = 3
2 * 7 and substitute

equation 13


4)
Write 32 and 16 as products of prime numbers 32=2
5 , 16=24 and substitute

equation 14




5)
Write 64 as products of prime numbers 64=2
6 and substitute

equation 15

Rationalize the denominator by multiplying numerator and denominator by (
3√7)2

equation 16


6)
Write 54 as products of prime numbers 54 = 2 * 3
3 and substitute

equation 17


7)
Multiply the denominator and numerator by the conjugate of the denominator

equation 18

Expand and simplify

equation 19



More Questions With Answers
    Use all the rules and properties of radicals to rationalize and simplify the following expressions.
    equation 20

Solutions to the Above Questions

    • Write 25 and 125 as the product of prime factors: 25 = 52 and 125 = 53, hence

      solution 1

    • Write 64 and 16 as the product of prime factors: 64=26 and 16 = 24, hence

      solution 2

    • Use product rule

      solution 3

    • Convert the mixed number under the radical into a fraction and substitute

      solution 42

      Use the division formula for radicals

      solution 43

      Write 64 and 27 as product of prime factors, substitute and simplify

      solution 44

    • Use the product formula and write 34 as the product of prime factors

      solution 51

      Simplify

      solution 52

      For √(17 x) and √(34 x) to be real numbers, x must be positive hence |x| = x

      solution 51

    • Write the radicand as a square and simplify

      solution 6

    • Write the radicand as the product of $2$ and a square and simplify

      solution 7

    • Simplify the radicand

      solution 81

      Write as the product of prime factors and simplify

      solution 82

    • Since n is a positive integer, then N=2n + 1 is an odd integer. Hence

      solution 9

    • Since n is a positive integer, then N=2n is an even integer. Hence

      solution 10

    • solution 11

    • Use division rule and simplify the radicand

      solution 12

    • Multiply numerator and denominator by the conjugate of the denominator

      solution 131

      Expand and simplify

      solution 132


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