If $2^x \times 8^{\frac{1}{5}} = 2^{\frac{1}{5}}$, then $x$ is equal to

### Concept & Formula The core concept is to convert all composite bases into prime bases so that the laws of indices can be applied uniformly. When dealing with bases like $2$ and $8$, recognize that $8 = 2^3$. The necessary laws of indices are: $$(x^m)^n = x^{mn}$$ $$x^m \times x^n = x^{m+n}$$ ### Step-by-Step Solution * **Given:** $$2^x \times 8^{\frac{1}{5}} = 2^{\frac{1}{5}}$$ * **Calculation:** Convert the base $8$ into a power of $2$. We know that $8 = 2^3$. Substitute $2^3$ into the equation: $$2^x \times (2^3)^{\frac{1}{5}} = 2^{\frac{1}{5}}$$ Apply the power of a power rule by multiplying the exponents: $$2^x \times 2^{\frac{3}{5}} = 2^{\frac{1}{5}}$$ Apply the product rule by adding the exponents on the left-hand side: $$2^{x + \frac{3}{5}} = 2^{\frac{1}{5}}$$ Since the bases on both sides are identical, equate the exponents: $$x + \frac{3}{5} = \frac{1}{5}$$ Isolate $x$ by subtracting $\frac{3}{5}$ from both sides: $$x = \frac{1}{5} - \frac{3}{5}$$ $$x = -\frac{2}{5}$$ ### Exam Strategy & Shortcut **Exponent-Only Equation:** Mentally translate $8$ to $2^3$ and skip writing the bases entirely. Immediately write down the linear equation for the exponents: $x + \frac{3}{5} = \frac{1}{5}$. Solving $x = \frac{1}{5} - \frac{3}{5} = -\frac{2}{5}$ takes less than 10 seconds and prevents transcription errors. ### Common Pitfall A frequent error is misapplying the product rule and multiplying the exponents instead of adding them, which leads to $x \cdot \frac{3}{5} = \frac{1}{5}$, incorrectly yielding $x = \frac{1}{3}$. Always remember that multiplying identical bases means adding their exponents. ### Final Answer **Therefore, the correct answer is -2/5.**