📌 Key Ideas & Notation
| Term | Meaning |
|---|---|
| x1⁄n | nth root of x, written √[n]{x}. Example: 91⁄2 = √9 = 3. |
| xm⁄n | First take the nth root, then raise to the power m: xm⁄n = (√[n]{x})m. Example: 82⁄3 = (³√8)2 = 22 = 4. |
| x−m⁄n | Take the reciprocal and then the fractional power: x−m⁄n = 1 ÷ xm⁄n. Example: 16−1⁄2 = 1 ÷ 161⁄2 = 1 ÷ 4. |
🌍 Real-World Connection:
Rational exponents appear whenever a law looks like y = kxm⁄n, for example in
physics (square-root dependence of period on length in pendulums) and
biology (body-mass scaling of metabolic rate).
Rational exponents appear whenever a law looks like y = kxm⁄n, for example in
physics (square-root dependence of period on length in pendulums) and
biology (body-mass scaling of metabolic rate).
📌 Why Write Roots as Powers?
Using rational exponents lets us apply the usual index laws (product, quotient, power of a power)
to expressions involving roots. This makes algebraic manipulation easier and more systematic.
🧠 Examiner Tip:
In AHL questions you are expected to:
In AHL questions you are expected to:
- Rewrite surds like √x and ³√x as x1⁄2 and x1⁄3.
- Use index laws to simplify, then convert back to root form only if the question asks.
📌 Interpreting a Rational Exponent
1. Denominator → root
- x1⁄2 means “square root of x”.
- x1⁄3 means “cube root of x”.
- In general, x1⁄n = √[n]{x}.
2. Numerator → power
- x3⁄2 = (√x)3.
- x5⁄3 = (³√x)5.
3. Negative sign → reciprocal
- x−1⁄2 = 1 ÷ x1⁄2 = 1 ÷ √x.
- x−3⁄4 = 1 ÷ x3⁄4 = 1 ÷ (⁴√x)3.
🔍 TOK Perspective:
Writing roots as powers is a choice of representation.
Does changing from √x to x1⁄2 change our understanding of the quantity,
or only our language for manipulating it?
Writing roots as powers is a choice of representation.
Does changing from √x to x1⁄2 change our understanding of the quantity,
or only our language for manipulating it?
📌 Using Index Laws with Rational Exponents
The usual index laws still hold for rational exponents (for x > 0):
- Product law: xa × xb = xa + b.
- Quotient law: xa ÷ xb = xa − b.
- Power of a power: (xa)b = xab.
Example 1 – Multiplying fractional powers
Simplify 51⁄2 × 51⁄3.
- Same base 5, so use the product law: add exponents.
- 1⁄2 + 1⁄3 = 3⁄6 + 2⁄6 = 5⁄6.
- Result: 51⁄2 × 51⁄3 = 55⁄6.
📐 IA Spotlight:
In an IA involving power-law data, you can write models as y = kxm⁄n.
Simplifying rational exponents makes it easier to compare different models and interpret
the meaning of the exponent m⁄n in your context.
In an IA involving power-law data, you can write models as y = kxm⁄n.
Simplifying rational exponents makes it easier to compare different models and interpret
the meaning of the exponent m⁄n in your context.
Example 2 – Power of a power (from the guide)
Simplify 323⁄5.
- Write 32 as a power of 2: 32 = 25.
- Now 323⁄5 = (25)3⁄5.
- Use the power-of-a-power rule: multiply exponents → 25 × 3⁄5 = 23.
- Final answer: 23 = 8.
Example 3 – Negative rational exponent
Show that x−1⁄2 = 1 ÷ √x.
- By definition of a negative exponent, x−1⁄2 = 1 ÷ x1⁄2.
- x1⁄2 is the square root of x, so x1⁄2 = √x.
- Therefore x−1⁄2 = 1 ÷ √x.
📱 GDC Tip:
When unsure, type both the original expression and your simplified answer into your GDC.
If the decimal values match, your simplification is very likely correct.
When unsure, type both the original expression and your simplified answer into your GDC.
If the decimal values match, your simplification is very likely correct.
📝 Paper 1 Strategy:
- First rewrite all roots as fractional powers (no radicals).
- Apply index laws carefully to combine or simplify.
- Only convert back to radical form if the question specifically wants a root.
❤️ CAS Idea:
Design a short workshop for younger students where you explain why 323⁄5 is exactly 8,
using both radical notation and rational exponents, and let them explore similar examples on a GDC.
Design a short workshop for younger students where you explain why 323⁄5 is exactly 8,
using both radical notation and rational exponents, and let them explore similar examples on a GDC.