Solve the equation $5^{(x − 2)} = (1 ÷ 125)^{(x +3)}$

Analysis

To solve this equation, we need to express both sides of the equation with the same base. Since the left side has a base of 5, we should aim to express the right side as a power of 5 as well.

Concept

The laws of indices are essential here:

1. Negative Exponents: $\frac{1}{a^n} = a^{-n}$
2. Power of a Power: $(a^m)^n = a^{m \times n}$
3. Equality of Bases: If $a^x = a^y$, then $x = y$ (provided $a > 0$ and $a \neq 1$).

Step-by-Step Solution

Step 1: Rewrite the right side of the equation. The equation is $5^{(x - 2)} = (\frac{1}{125})^{(x + 3)}$. We know that $125 = 5^3$. Therefore, $\frac{1}{125} = \frac{1}{5^3} = 5^{-3}$.

Step 2: Substitute this back into the equation. $5^{(x - 2)} = (5^{-3})^{(x + 3)}$

Step 3: Simplify the exponents on the right side. Using the power of a power rule $(a^m)^n = a^{mn}$: $5^{(x - 2)} = 5^{-3(x + 3)}$ $5^{(x - 2)} = 5^{(-3x - 9)}$

Step 4: Equate the exponents. Since the bases are now the same, we can equate the exponents: $x - 2 = -3x - 9$

Step 5: Solve for $x$. Add $3x$ to both sides: $x + 3x - 2 = -9$ $4x - 2 = -9$

Add 2 to both sides: $4x = -9 + 2$ $4x = -7$

Divide by 4: $x = -\frac{7}{4}$

Conclusion

Comparing our result to the given options, we find that $x = -\frac{7}{4}$ matches option B.

Exam Tip: Whenever you see fractions like $1/25, 1/125,$ or $1/625$ in an exponential equation, immediately think of negative powers of 5. It will save you time during the exam!