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IB Mathematics AI SL Simplifying expressions Study Notes - New Syllabus

IB Mathematics AI SL Simplifying expressions Study Notes

LEARNING OBJECTIVE

  • Simplifying expressions, both numerically and algebraically, involving rational exponents.

Key Concepts: 

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MAI HL and SL Notes – All topics

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

1. \( x^{\frac{1}{6}} = ?\)
2. \( m^{\frac{2}{3}} = ?\)
3. \(\sqrt{m^2} = ?\)

▶️Answer/Explanation

Solution:

1. \( x^{\frac{1}{6}} = \sqrt[6]{x} \)
2. \( m^{\frac{2}{3}} = \sqrt[3]{m^2} \)
3. \(\sqrt{m^2} = m^{\frac{2}{2}} = m^1 = m\)

♦  Exponential Rules Applied to Rational Exponents
The standard exponent rules apply equally to rational exponents:

♦ 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

\( \frac{8x^{\frac{3}{4}}y^2}{2x^{\frac{1}{4}}y^{\frac{2}{3}}}=?\)

▶️Answer/Explanation

Solution:

\( \frac{8x^{\frac{3}{4}}y^2}{2x^{\frac{1}{4}}y^{\frac{2}{3}}} = 4x^{\frac{3}{4}-\frac{1}{4}}y^{2-\frac{2}{3}} = 4x^{\frac{1}{2}}y^{\frac{4}{3}} \)

 

Evaluating Expressions

Evaluate \( (-125)^{\frac{2}{3}} \)

▶️Answer/Explanation

Solution:

1. Convert to radical form: \( (\sqrt[3]{-125})^2 \)
2. Evaluate cube root: \( (-5)^2 \)
3. Square result: 25

 

Negative Exponents

Evaluate \( \left(\frac{36}{49}\right)^{-\frac{3}{2}} \)

▶️Answer/Explanation

Solution:

1. Apply negative exponent: \( \left(\frac{49}{36}\right)^{\frac{3}{2}} \)
2. Take square root: \( \left(\frac{7}{6}\right)^3 \)
3. Cube result: \( \frac{343}{216} \)

 

Combining Terms

Simplify \( \left( 3a^{\frac{3}{2}} \right) \left( -7a^{\frac{1}{5}} \right) \)

▶️Answer/Explanation

Solution:

1. Multiply coefficients: \( 3 \times -7 = -21 \)
2. Add exponents: \( a^{\frac{3}{2}+\frac{1}{5}} = a^{\frac{17}{10}} \)
3. Final answer: \( -21a^{\frac{17}{10}} \)

 

Complex Simplification

Simplify \( \frac{x^{2a-3} \cdot x^{-a+1}}{(x^2)^{a-4}} \)

▶️Answer/Explanation

Solution:

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

Consider simplifying \( \frac{3}{\sqrt[3]{3}} \):

▶️Answer/Explanation

Solution:

Method 1: Rationalizing
\( \frac{3}{\sqrt[3]{3}} \times \frac{\sqrt[3]{3^2}}{\sqrt[3]{3^2}} = \frac{3\sqrt[3]{9}}{3} = \sqrt[3]{9} \)

Method 2: Rational Exponents
\(\frac{3}{3^{\frac{1}{3}}} = 3^{1 – \frac{1}{3}} = 3^{\frac{2}{3}} = \sqrt[3]{9}\)

♦Simplifying Compound Expressions

Example

Simplify \( \frac{y^{\frac{1}{2}} + 1}{y^{\frac{1}{2}} – 1} \times \frac{y^{\frac{1}{2}} + 1}{y^{\frac{1}{2}} + 1} \)

▶️Answer/Explanation

Solution:

Multiply by conjugate:
\( \frac{(y^{\frac{1}{2}} + 1)^2}{y – 1} = \frac{y + 2y^{\frac{1}{2}} + 1}{y – 1} \)

 

♦ 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

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