Grade 9 · Grade 9: Radicals, Rational Exponents and Quantities · Radicals, Indices and Precision
Radicals, Rational Exponents and Precision
Simplify surds, move between radical and index form, and report answers to a sensible accuracy.
Learning objectives
- Simplify and operate with radical expressions, including rationalising simple denominators.
- Rewrite radicals using rational exponents.
- Use units to guide multi-step solutions.
- Calculate percent error and choose an appropriate accuracy.
AERO Mathematics alignment
AERO.M9.NS.1
Simplify and perform operations with radical expressions, including rationalising simple denominators.
AERO.M9.NS.2
Rewrite expressions involving radicals using rational exponents and explain the equivalence.
AERO.M9.NS.3
Use units to guide the solution of multi-step problems, choose levels of accuracy appropriate to limitations of measurement, and calculate percent error.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Two students calculate the diagonal of a 1 m square. One writes √2 m; the other writes 1.4 m.
Think about it
Which answer is correct, and when does the difference matter?
Hint: Think about what happens after ten more calculations.
EXPLAIN
Surds, indices and accuracy
Simplifying surds
Split out perfect squares: √50 = √(25 × 2) = 5√2. Only like surds add: 5√2 + 3√2 = 8√2, but 5√2 + 3√3 cannot be combined.
Rationalising a denominator
Multiply numerator and denominator by the surd: 3/√2 = 3√2/2. The value is unchanged because you multiplied by 1.
Rational exponents
| Radical form | Index form | Value |
|---|---|---|
| √x | x^(1/2) | √9 = 3 |
| ∛x | x^(1/3) | ∛27 = 3 |
| ∛(x²) | x^(2/3) | 8^(2/3) = 4 |
Accuracy and error
Report answers to an accuracy justified by your measurements. Percent error = |measured − actual| ÷ actual × 100. Rounding early accumulates error, so carry exact values until the final step.
Remember
√2 is exact; 1.41 is an approximation. Exact form belongs in algebra and proof, and decimals belong in the final practical answer.
INVESTIGATION
Try it yourself
Calculate the diagonal of a 1 m square ten times in a chain, once using √2 throughout and once rounding to 1.4 at every step. Compare the two totals and calculate the percent error introduced by rounding.
Watch
Laws of Exponents
Math Antics · 8:00
Index laws underpin every rational-exponent conversion in this lesson.
Before you watch: What does x^(1/2) mean?
- Simplify √50 + √18.
- Write ∛(x²) using a rational exponent.
Interactive simulation · PhET
Number Line: Distance
Compare exact surd positions with rounded decimal approximations.
While you explore
- How much error does rounding √3 to 1.7 introduce?
- When is exact form essential?
Key vocabulary
- Rationalising
- Removing a root from the bottom of a fraction.
- Surd
- A root that cannot be simplified to a whole number.
Practice questions
0/1 correct
Level 1 · Criterion A
Simplify √72.
MYP criterion tasks
Level 2 · Criterion C
Rationalise 5/√3 and explain why the value does not change.
Reflect & track
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