IB MYP 4-5 Maths-Laws of exponents, fractional/rational exponents - Study Notes - New Syllabus
IB MYP 4-5 Maths- Laws of exponents, fractional/rational exponents – Study Notes
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- Laws of exponents, fractional/rational exponents
IB MYP 4-5 Maths- Laws of exponents, fractional/rational exponents – Study Notes – All topics
Laws of Exponents: Fractional (Rational) Exponents
Laws of Exponents: Fractional (Rational) Exponents
Fractional or rational exponents are another way to represent roots. They follow all the same exponent rules as integer exponents.
Concept:
A rational exponent of the form \( a^{\frac{m}{n}} \) means:
\( a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m \)
For example, \( 8^{\frac{2}{3}} = \sqrt[3]{8^2} = \left( \sqrt[3]{8} \right)^2 = 2^2 = 4 \)
Important Laws with Fractional Exponents:
| Rule | Example |
|---|---|
| \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \) | \( 16^{\frac{3}{4}} = \sqrt[4]{16^3} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8 \) |
| \( a^r \cdot a^s = a^{r + s} \) | \( 9^{\frac{1}{2}} \cdot 9^{\frac{3}{2}} = 9^2 = 81 \) |
| \( \frac{a^r}{a^s} = a^{r – s} \) | \( \frac{27^{\frac{2}{3}}}{27^{\frac{1}{3}}} = 27^{\frac{1}{3}} = 3 \) |
| \( (a^r)^s = a^{r \cdot s} \) | \( \left( 4^{\frac{1}{2}} \right)^2 = 4^{1} = 4 \) |
Example: Simplify \( 81^{\frac{3}{4}} \)
▶️ Answer/Explanation
\( 81^{\frac{3}{4}} = \sqrt[4]{81^3} = \left( \sqrt[4]{81} \right)^3 = 3^3 = \boxed{27} \)
Example : Evaluate \( 32^{\frac{2}{5}} \)
▶️ Answer/Explanation
\( 32^{\frac{2}{5}} = \sqrt[5]{32^2} = \left( \sqrt[5]{32} \right)^2 = 2^2 = \boxed{4} \)
Example : Simplify \( \frac{64^{\frac{2}{3}}}{8^{\frac{2}{3}}} \)
▶️ Answer/Explanation
Use quotient rule: \( \frac{64^{\frac{2}{3}}}{8^{\frac{2}{3}}} = \left( \frac{64}{8} \right)^{\frac{2}{3}} = 8^{\frac{2}{3}} \)
\( = \sqrt[3]{8^2} = \sqrt[3]{64} = \boxed{4} \)
Example : Simplify \( (16^{\frac{1}{2}})^4 \)
▶️ Answer/Explanation
\( (16^{\frac{1}{2}})^4 = 16^{\frac{4}{2}} = 16^2 = \boxed{256} \)