Solving Equation of Index Number – Example 1

$$\eqalign{
& {5^{x – 1}} + {5^{x + 2}} = 3150 \cr
& \frac{{{5^x}}}{5} + {5^x} \times {5^2} = 3150 \cr
& \frac{{{5^x}}}{5} + \frac{{125 \times {5^x}}}{5} = 3150 \cr
& 126 \times {5^x} = 5 \times 3150 \cr
& {5^x} = \frac{{5 \times 3150}}{{126}} \cr
& {5^x} = 125 = {5^3} \cr
& x = 3 \cr} $$

$$\eqalign{
& {a^m} \times {a^n} = {a^{m + n}} \cr
& {a^m} \div {a^n} = \frac{{{a^m}}}{{{a^n}}} = {a^{m – n}} \cr
& {\text{Hence}} \cr
& {5^{x – 1}} = {5^x} \div 5 = \frac{{{5^x}}}{5} \cr
& {5^{x + 2}} = {5^x} \times {5^2} \cr} $$