The value of $5^{\frac{1}{4}} \times (125)^{0.25}$ is :

### Concept & Formula This problem tests the application of index laws by combining terms with identical bases. It also requires converting decimal exponents into fractional exponents to standardize the expression for evaluation. $$ a^m \times a^n = a^{m+n} $$ $$ (a^m)^n = a^{mn} $$ ### Step-by-Step Solution * **Given:** The expression to evaluate is $5^{\frac{1}{4}} \times (125)^{0.25}$. * **Calculation:** 1. Convert the decimal exponent to a fraction for algebraic consistency: $0.25 = \frac{25}{100} = \frac{1}{4}$ The expression becomes $5^{\frac{1}{4}} \times (125)^{\frac{1}{4}}$. 2. Convert the composite base ($125$) into a prime base of $5$: $125 = 5^3$ 3. Substitute this back into the expression: $5^{\frac{1}{4}} \times (5^3)^{\frac{1}{4}}$ 4. Apply the power of a power rule $(x^a)^b = x^{ab}$ to the second term: $5^{\frac{1}{4}} \times 5^{\frac{3}{4}}$ 5. Since the bases are identical, apply the product rule by adding the exponents: $5^{(\frac{1}{4} + \frac{3}{4})}$ 6. Simplify the exponent fraction: $\frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1$ 7. Calculate the final simplified value: $5^1 = 5$ ### Exam Strategy & Shortcut Instead of standardizing bases first, you can use the power of a product rule in reverse: $a^n \times b^n = (a \times b)^n$. Notice that both terms effectively have an exponent of $1/4$ (since $0.25 = 1/4$). You can immediately rewrite the expression as $(5 \times 125)^{\frac{1}{4}}$. Multiply the bases together to get $625^{\frac{1}{4}}$. Since you should have standard powers memorized ($625 = 5^4$), the 4th root of $625$ is simply $5$. This shortcut skips fractional exponent addition entirely and saves critical time. ### Common Pitfall A frequent mistake is attempting to evaluate $(125)^{0.25}$ independently without noticing its relationship to the adjacent term. Finding the 4th root of $125$ yields a messy irrational number, causing students to either waste time attempting to estimate it or guess incorrectly, rather than leveraging the exponent laws to combine the terms. ### Final Answer **Therefore, the correct answer is 5.**