Solving Exponential (Indicial) Equations
Solving Exponential (Indicial) Equations
Grade 11 Mathematics • G.C.E. O/L Sri Lanka Local Syllabus
Learning Objectives
- Solve exponential equations where the unknown variable is in the exponent by equating prime bases.
- Solve indicial equations involving fractions, roots, and multiple exponential terms.
- Solve power equations where the unknown variable is the base raised to a fractional exponent.
Two Core Equation Methods
Step-by-Step Worked Examples
- Express both sides in prime base \(2\) (\(4 = 2^2\) and \(\frac{1}{8} = \frac{1}{2^3} = 2^{-3}\)): \[(2^2)^{2x-1} = 2^{-3} \implies 2^{4x-2} = 2^{-3}\]
- Since the bases are equal, equate the exponents: \[4x - 2 = -3\]
- Solve the linear equation for \(x\): \[4x = -1 \implies x = -\frac{1}{4}\]
- Isolate the term containing \(x\) by dividing both sides by \(2\): \[x^{\frac{2}{3}} = 9\]
- Raise both sides to the reciprocal power \(\frac{3}{2}\): \[\left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 9^{\frac{3}{2}} \implies x = (3^2)^{\frac{3}{2}}\]
- Simplify the right-hand side: \[x = 3^3 = 27\]
Practice Drill (O/L Exam Style)
Show Step-by-Step Answer
Step 1: Convert \(9\) and \(27\) to base \(3\):
\(3^{x+2} \times (3^2)^x = 3^3 \implies 3^{x+2} \times 3^{2x} = 3^3\)
Step 2: Apply the product law on the left-hand side:
\(3^{3x+2} = 3^3\)
Step 3: Equate exponents and solve:
\(3x + 2 = 3 \implies 3x = 1 \implies \mathbf{x = \frac{1}{3}}\)
Show Step-by-Step Answer
Step 1: Rewrite \(\sqrt{2^x}\) as \(2^{\frac{x}{2}}\) and \(16\) as \(2^4\):
\(2^{\frac{x}{2}} = 2^4\)
Step 2: Equate exponents:
\(\frac{x}{2} = 4 \implies \mathbf{x = 8}\)
Show Step-by-Step Answer
Step 1: Rewrite both sides with positive exponents:
\(\frac{1}{x^{\frac{3}{2}}} = \frac{1}{8} \implies x^{\frac{3}{2}} = 8\)
Step 2: Raise both sides to the reciprocal power \(\frac{2}{3}\):
\(x = 8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^2 = \mathbf{4}\)