1. **Problem a:** Solve the indicial equation $$2^x + 2^{x-1} = 48$$ Step 1: Rewrite the terms to factorize: $$2^x + 2^{x-1} = 2^x + \frac{2^x}{2} = 2^x + \frac{2^x}{2} = 2^x\left(1 + \frac{1}{2}\right) = 2^x \times \frac{3}{2}$$ Step 2: Set the equation: $$2^x \times \frac{3}{2} = 48$$ Step 3: Multiply both sides by \(\frac{2}{3}\): $$2^x = 48 \times \frac{2}{3} = 32$$ Step 4: Recognize 32 as a power of 2: $$32 = 2^5$$ Step 5: Equate powers: $$2^x = 2^5 \implies x = 5$$ 2. **Problem b:** Solve $$ (2x+3)^3 = 125 $$ Step 1: Take cube root on both sides: $$ 2x+3 = \sqrt[3]{125} = 5 $$ Step 2: Solve for \(x\): $$ 2x = 5 - 3 = 2 $$ $$ x = \frac{2}{2} = 1 $$ 3. **Problem c:** Solve $$ 0.2x = \frac{1}{25} $$ Step 1: Convert decimals to fraction: $$ 0.2 = \frac{1}{5} $$ Step 2: Rewrite equation: $$ \frac{1}{5} x = \frac{1}{25} $$ Step 3: Multiply both sides by 5: $$ x = \frac{1}{25} \times 5 = \frac{5}{25} = \frac{1}{5} $$ 4. **Problem d:** Solve $$ 3x^{3/5} = 81 $$ Step 1: Divide both sides by 3: $$ x^{3/5} = \frac{81}{3} = 27 $$ Step 2: Write 27 as power of 3: $$ 27 = 3^3 $$ Step 3: So $$ x^{3/5} = 3^3 $$ Step 4: Raise both sides to the power \(\frac{5}{3}\): $$ \left(x^{3/5}\right)^{5/3} = \left(3^3\right)^{5/3} $$ $$ x = 3^{3 \times \frac{5}{3}} = 3^5 = 243 $$ 5. **Evaluate expression a:** $$ 125^{-1/3} \times 25^{1/2} \div 49^{-1/2} \times 7 $$ Step 1: Rewrite bases as powers: $$ 125 = 5^3, \quad 25 = 5^2, \quad 49 = 7^2 $$ Step 2: Substitute and apply exponents: $$ (5^3)^{-1/3} \times (5^2)^{1/2} \div (7^2)^{-1/2} \times 7 $$ Step 3: Simplify exponents: $$ 5^{3 \times (-1/3)} \times 5^{2 \times (1/2)} \div 7^{2 \times (-1/2)} \times 7 $$ $$ = 5^{-1} \times 5^{1} \div 7^{-1} \times 7 $$ Step 4: Simplify multiplication and division: $$ 5^{-1+1} \times 7^{1+1} = 5^0 \times 7^2 = 1 \times 49 = 49 $$ 6. **Evaluate expression b:** $$ \frac{1}{4}(2^n - 2^{n \times \text{th}}) $$ Step 1: The problem statement is unclear for the exponent \(n \times \text{th}\). Please clarify the meaning of \(\text{th}\) or provide correct notation to proceed. **Final answers:** - (a) \(x = 5\) - (b) \(x = 1\) - (c) \(x = \frac{1}{5}\) - (d) \(x = 243\) - (Evaluate a) = 49 - (Evaluate b) = Cannot determine without clarification