IB Mathematics AI AHL Simplifying expressions Study Notes - New Syllabus
IB Mathematics AI AHL Simplifying expressions Study Notes
LEARNING OBJECTIVE
- Simplifying expressions, both numerically and algebraically, involving rational exponents.
Key Concepts:
- Simplifying expressions
- IBDP Maths AI SL- IB Style Practice Questions with Answer-Topic Wise-Paper 1
- IBDP Maths AI SL- IB Style Practice Questions with Answer-Topic Wise-Paper 2
- IB DP Maths AI HL- IB Style Practice Questions with Answer-Topic Wise-Paper 1
- IB DP Maths AI HL- IB Style Practice Questions with Answer-Topic Wise-Paper 2
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Introduction to Rational Exponents
♦ Rational exponents provide an alternative way to express radicals and roots. The general form is:
\( b^{\frac{m}{n}} = \sqrt[n]{b^m} = (\sqrt[n]{b})^m \)
where:
\( n \) is the index of the radical (denominator of the exponent)
\( m \) is the power of the radicand (numerator of the exponent)
♦ Key Concepts and Rules
1. Conversion Between Radical and Exponential Forms
\(\sqrt[n]{b} = b^{\frac{1}{n}}\)
\(\sqrt[n]{b^m} = b^{\frac{m}{n}}\)
Example Solve the following 1. \( x^{\frac{1}{6}} = ?\) ▶️Answer/ExplanationSolution: 1. \( x^{\frac{1}{6}} = \sqrt[6]{x} \) |
♦ Exponential Rules Applied to Rational Exponents
The standard exponent rules apply equally to rational exponents:
| Rule | Example |
| $x^a \cdot x^b = x^{a + b}$ | $\sqrt[3]{x^5} \cdot \sqrt[4]{x^4} = x^{5/3} \cdot x^{1} = x^3$ |
| $\frac{x^a}{x^b} = x^{a – b}$ | $\frac{\sqrt[4]{x^3}}{\sqrt[4]{x}} = x^{3/4} \div x^{1/4} = x^{2/4} = x^{1/2}$ |
| $(x^a)^b = x^{ab}$ | $(\sqrt{x})^4 = (x^{1/2})^4 = x^2$ |
| $x^{-a} = \frac{1}{x^a}$ | $49^{-1/2} = \frac{1}{7}$ |
| $(xy)^a = x^a \cdot y^a$ | $(6x^5y^7)^{1/3} = 6^{1/3} x^{5/3} y^{7/3}$ |
♦ Simplification Techniques
An expression with rational exponents is considered simplified when:
1. No negative exponents remain
2. No fractional exponents are in denominators
3. No complex fractions exist
4. Radical indices are minimized
Example Simplify the given expression \( \frac{8x^{\frac{3}{4}}y^2}{2x^{\frac{1}{4}}y^{\frac{2}{3}}}=?\) Show the steps. ▶️Answer/ExplanationSolution: \( \frac{8x^{\frac{3}{4}}y^2}{2x^{\frac{1}{4}}y^{\frac{2}{3}}} = 4x^{\frac{3}{4}-\frac{1}{4}}y^{2-\frac{2}{3}} = 4x^{\frac{1}{2}}y^{\frac{4}{3}} \) |
Evaluating Expressions Evaluate \( (-125)^{\frac{2}{3}} \) Show the steps. ▶️Answer/ExplanationSolution: 1. Convert to radical form: \( (\sqrt[3]{-125})^2 \) |
Negative Exponents Evaluate \( \left(\frac{36}{49}\right)^{-\frac{3}{2}} \) Show the Steps. ▶️Answer/ExplanationSolution: 1. Apply negative exponent: \( \left(\frac{49}{36}\right)^{\frac{3}{2}} \) |
Combining Terms Simplify \( \left( 3a^{\frac{3}{2}} \right) \left( -7a^{\frac{1}{5}} \right) \) Show the Steps. ▶️Answer/ExplanationSolution: 1. Multiply coefficients: \( 3 \times -7 = -21 \) |
Complex Simplification Simplify \( \frac{x^{2a-3} \cdot x^{-a+1}}{(x^2)^{a-4}} \) Show the Steps. ▶️Answer/ExplanationSolution: 1. Combine numerator: \(x^{(2a – 3) + (-a + 1)} = x^{a – 2}\) 2. Simplify denominator: \(x^{2(a – 4)} = x^{2a – 8}\) 3. Divide: \(x^{(a – 2) – (2a – 8)} = x^{-a + 6}\) |
♦ Rationalizing vs. Rational Exponents
Example Simplify \( \frac{3}{\sqrt[3]{3}} \): Show the Steps. ▶️Answer/ExplanationSolution: Method 1: Rationalizing Method 2: Rational Exponents |
♦Simplifying Compound Expressions
Example Simplify \( \frac{y^{\frac{1}{2}} + 1}{y^{\frac{1}{2}} – 1} \times \frac{y^{\frac{1}{2}} + 1}{y^{\frac{1}{2}} + 1} \) Show the Steps. ▶️Answer/ExplanationSolution: Multiply by conjugate: |
♦ Tips
1. When adding exponents, find a common denominator
2. Remember that \( x^{\frac{a}{b}} \) is not the same as \( \frac{x^a}{b} \)
3. Negative exponents indicate reciprocals, not negative values
4. Always reduce fractional exponents to simplest form
