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

Examples

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Roots of Real Numbers and Radicals .
2) Product (Multiplication) formula of radicals with equal indices is given by

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

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Example

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5) You may rewrite expressions without radicals (to rationalize denominators) as follows
A) Example 1:

B) Example 2:

C) Example 3:

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Rationalize Denominators of Radical Expressions .

Examples
Rationalize and simplify the given expressions

1)
Write 128 and 32 as product/powers of prime factors: 128 = 2
7 , 32 = 2 5 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 = 3
2 × 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

Use all the rules and properties of radicals to rationalize and simplify the following expressions.

Solutions to the Above Questions

1. Write 25 and 125 as the product of prime factors: 25 = 52 and 125 = 53, hence
2. Write 64 and 16 as the product of prime factors: 64 = 26 and 16 = 24, hence
3. Use product rule
4. 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
5. 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
6. Write the radicand as a square and simplify
7. Write the radicand as the product of $2$ and a square and simplify

Write as the product of prime factors and simplify
9. Since n is a positive integer, then N = 2 n + 1 is an odd integer. Hence
10. Since n is a positive integer, then N = 2 n is an even integer. Hence
11. Use division rule and simplify the radicand
12. Multiply numerator and denominator by the conjugate of the denominator

Expand and simplify

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