Practice Maths

Operations with Real Numbers

Key Ideas

Key Terms

Like surds
Have the same irrational part: 3√2 and 5√2 are like surds (can be added/subtracted).
Adding/subtracting
Only like surds can be combined: 3√2 + 5√2 = 8√2.
Product rule
√a × √b = √(ab)   (a, b ≥ 0).
Quotient rule
√a ÷ √b = √(a/b)   (a, b ≥ 0, b ≠ 0).
Coefficients multiply separately
A√m × b√n = ab√(mn), then simplify.
OperationRule / Example
Add / Subtract3√2 + 5√2 = 8√2   (like surds only)
Multiply√a × √b = √(ab)   e.g. √3 × √5 = √15
Divide√a ÷ √b = √(a/b)   e.g. √18 ÷ √2 = √9 = 3
Coefficients2√3 × 4√5 = 8√15
Simplify first√8 + √18 = 2√2 + 3√2 = 5√2
Hot Tip Before adding or subtracting surds, always simplify each one first. √8 + √2 cannot be combined until you write √8 = 2√2; then 2√2 + √2 = 3√2.

Worked Example — Adding Like Surds

Question: Simplify 3√12 + √75 − 2√3.

Step 1: Simplify each surd: 3√12 = 3×2√3 = 6√3; √75 = 5√3.

Step 2: Collect like surds: 6√3 + 5√3 − 2√3 = 9√3.

Worked Example — Multiplying Surds

Question: Simplify 3√6 × 2√10.

3√6 × 2√10 = 6√60 = 6√(4 × 15) = 6 × 2√15 = 12√15.

Rules for Combining Rational and Irrational Numbers

When you add, subtract, multiply, or divide real numbers, the result may be rational or irrational depending on the types you start with. Here are the reliable rules:

Rational ± Rational = Rational. For example, 3/4 + 1/2 = 5/4. Always rational.

Rational ± Irrational = Irrational. For example, 2 + √3 is irrational because you cannot simplify it any further into a fraction.

Rational × Rational = Rational (as long as neither is zero).

Rational × Irrational = Irrational (usually, unless the rational number is 0). For example, 5 × √2 = 5√2, which is irrational.

Irrational × Irrational = can be either. √2 × √2 = 2 (rational!). But √2 × √3 = √6 (irrational).

Adding and Subtracting Real Numbers

Adding and subtracting works the same way as with regular numbers, but you can only combine like surds (irrational terms with the same radical part). Think of it like collecting like terms in algebra.

For example: 3√2 + 5√2 = 8√2. But 3√2 + 5√3 cannot be simplified — they are different types, like 3x + 5y in algebra.

You can always add rational numbers separately from irrational ones: (2 + 3√5) + (4 + √5) = 6 + 4√5.

Multiplying and Dividing Real Numbers

Use the product rule: √a × √b = √(a × b). So √3 × √7 = √21. And √5 × √5 = √25 = 5.

Use the quotient rule: √a ÷ √b = √(a/b). So √50 ÷ √2 = √25 = 5.

When multiplying expressions like 3√2 × 4√5, multiply the whole-number parts together and the surd parts together: (3 × 4) × (√2 × √5) = 12√10.

Order of Operations with Real Numbers

The order of operations (BODMAS/BIDMAS) applies to all real numbers, including irrationals. Brackets first, then powers/roots, then multiplication and division left to right, then addition and subtraction left to right.

Example: 2 + 3 × √4 = 2 + 3 × 2 = 2 + 6 = 8. Do not add the 2 and 3 first — multiplication comes before addition.

Example: (1 + √5)(1 − √5) = 1 − √5 + √5 − 5 = 1 − 5 = −4. The middle terms cancel, giving a rational result from two irrational numbers.

Key tip: The statement "√2 × √2 = 2" is one of the most useful facts in this topic. Whenever you see a surd squared, it becomes rational. This is why (3 + √2)(3 − √2) = 9 − 2 = 7 — it is the difference of two squares pattern applied to surds, and the result is always rational.

Mastery Practice

  1. Add or subtract, simplifying all surds first. Fluency

      Expression Simplified Answer
    (a)3√2 + 5√2 
    (b)7√3 − 2√3 
    (c)√8 + √18 
    (d)√12 + √27 
    (e)2√50 − √8 
    (f)√75 + √48 − √27 
    (g)3√20 − √45 + 2√5 
    (h)2√72 + 3√32 − √50 

  2. Multiply or divide and simplify fully. Fluency

      Expression Simplified Answer
    (a)√3 × √3 
    (b)√5 × √20 
    (c)2√3 × 4√3 
    (d)3√2 × 5√8 
    (e)√6 × √10 
    (f)√50 ÷ √2 
    (g)√75 ÷ √3 
    (h)6√20 ÷ 3√5 

  3. Simplify each mixed-operation expression. Fluency

      Expression Simplified Answer
    (a)(√7)² 
    (b)(3√2)² 
    (c)2√5 × 3√15 
    (d)(√3 + √3)² 
    (e)√2 × √8 + √3 × √3 
    (f)4√6 ÷ (2√2) 

  4. State whether each claim is True or False, with justification. Fluency

    1. √2 + √3 = √5
    2. √4 × √9 = √36 = 6
    3. 3√2 + 4√3 = 7√5
    4. √a × √a = a for any a ≥ 0
    5. √50 − √2 = √48
    6. 2√3 is the same as √12

  5. Surd perimeters. Understanding

    Garden Border. A rectangular garden has length √50 m and width √8 m.
    1. Simplify √50 and √8 into simplest surd form.
    2. Find the exact perimeter in simplest surd form.
    3. Find the exact area. Explain why the area is rational even though both side lengths are irrational.

  6. Tiling a floor. Understanding

    Square Tiles. A square tile has a side length of √3 m. Six tiles are placed in a row.
    1. Write an expression for the total length of six tiles in a row.
    2. Simplify this expression.
    3. What is the area of one tile? Is this rational or irrational?

  7. Comparing surd expressions. Understanding

    Which is larger? Use the squaring technique or decimal estimates to compare each pair.
    1. 3√2 versus 2√3
    2. 4√3 versus 3√5
    3. √8 + √2 versus √18

  8. Identify whether each result is rational or irrational, without a calculator. Understanding

      Expression Rational or Irrational? Value / Reason
    (a)√2 × √2  
    (b)√2 + √2  
    (c)√3 × √12  
    (d)√5 + 1  
    (e)(√5)² − 5  

  9. Pythagoras’ theorem with surds. Problem Solving

    Right Triangles. Use Pythagoras’ theorem (c² = a² + b²) and surd arithmetic to find exact side lengths.
    1. A right triangle has legs √5 cm and √20 cm. Find the exact hypotenuse length.
    2. A right triangle has hypotenuse √13 cm and one leg √4 cm. Find the exact other leg.
    3. A square has diagonal √72 cm. Find the exact side length. (Hint: diagonal of square = s√2.)

  10. Surd investigation — patterns in multiplication. Problem Solving

    Pattern Investigation. Explore what happens when you multiply and add surds in different combinations.
    1. Evaluate (√a + √b)(√a − √b) for a = 5, b = 3. Then expand using algebra to find a general rule.
    2. Use your rule to evaluate (√7 + √2)(√7 − √2) without a calculator.
    3. A student claims that √a + √b = √(a+b) for all positive a and b. Disprove this with a specific counter-example and explain the error.
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