Square Roots of Non-squares
Approximate or simplify surds like √50 = 5√2.
Key idea
Approximate or simplify surds like √50 = 5√2.
In depth
If n is not a perfect square, √n is irrational, but you can trap it between neighbouring perfect squares and refine by decimals. Leave answers in surd form (like 2√3) for exactness.
Worked example
√20: between 4² = 16 and 5² = 25, so roughly 4.47. Exactly √20 = √4 × √5 = 2√5.
Formula
Real-world uses
- Finding the exact diagonal of a rectangular screen or room
- Simplifying expressions in construction
Common mistake
Rounding too soon in multi-step problems.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Square Roots of Non-squares?
Approximate or simplify surds like √50 = 5√2. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Square Roots of Non-squares?
The key formula is √(ab) = √a·√b.
Where is Square Roots of Non-squares used in real life?
Real-world examples: Finding the exact diagonal of a rectangular screen or room; Simplifying expressions in construction.
What should I learn before Square Roots of Non-squares?
Make sure you are comfortable with Real Numbers, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Square Roots of Non-squares?
Rounding too soon in multi-step problems.
How can I practise Square Roots of Non-squares?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Square Roots of Non-squares?
Move on to Rationalizing Denominators next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.