Simplify Radicals
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 \( n^{\text{th}} \) 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.

Questions

Rationalize and simplify the given expressions equation 10

Answers to the above questions

1) Write 128 and 32 as products of prime factors: \( 128 = 2^7 \), \( 32 = 2^5 \) hence

equation 11

2) Use product rule to write that \( \sqrt{2} \cdot \sqrt{6} = \sqrt{12} \)

equation 12

3) Write 14 and 63 as products of prime numbers \( 14 = 2 \times 7 \) , \( 63 = 3^2 \times 7 \) and substitute

equation 13

4) Write 32 and 16 as products of prime numbers \( 32 = 2^5\) , \( 16 = 2^4 \) 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 \( \left( \sqrt[3]{7} \right)^2 \)

equation 16

6) Write 54 as products of prime numbers \( 54 = 2 \times 3^3 \) 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

  1. Write 25 and 125 as the product of prime factors: \(25 = 5^2 \) and \( 125 = 5^3 \), hence
    solution 1
  2. Write 64 and 16 as the product of prime factors: \( 64 = 2^6 \) and \( 16 = 2^4 \), hence
    solution 2
  3. Use product rule solution 3
  4. 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

  5. Use the product formula and write 34 as the product of prime factors

    solution 51

    Simplify

    solution 52

    For \( \sqrt{17x} \) and \( \sqrt{34x} \) to be real numbers, \( x \) must be positive, hence \( |x| = x \).

    solution 51

  6. Write the radicand as a square and simplify

    solution 6

  7. Write the radicand as the product of \( 2 \) and a square and simplify

    solution 7

  8. Simplify the radicand

    solution 81

    Write as the product of prime factors and simplify solution 82

  9. Since \( n \) is a positive integer, then \( N = 2 n + 1 \) is an odd integer. Hence

    solution 9

  10. Since \( n \) is a positive integer, then \( N = 2 n \) is an even integer. Hence

    solution 10

  11. solution 11
  12. Use division rule and simplify the radicand

    solution 12

  13. Multiply numerator and denominator by the conjugate of the denominator

    solution 131

    Expand and simplify

    solution 132

Links and References