Algebra 1 Question of the Day

1. $x^{\frac{5}{4}}$
2. $x^{\frac{7}{4}}$ (correct answer)
3. $x^{\frac{3}{2}}$
4. $x^{\frac{13}{4}}$

Explanation: When you see radical expressions multiplied together, you're working with exponent rules in disguise. The key insight is converting radicals to fractional exponents, then using the rule that when multiplying powers with the same base, you add the exponents. First, convert each radical to exponential form. Remember that $\sqrt{x^3} = (x^3)^{\frac{1}{2}}$ and $\sqrt[4]{x} = x^{\frac{1}{4}}$. Using the power rule $(a^m)^n = a^{mn}$, we get $\sqrt{x^3} = x^{3 \cdot \frac{1}{2}} = x^{\frac{3}{2}}$. Now multiply: $x^{\frac{3}{2}} \cdot x^{\frac{1}{4}}$. When multiplying powers with the same base, add exponents: $\frac{3}{2} + \frac{1}{4}$. To add these fractions, find a common denominator: $\frac{3}{2} = \frac{6}{4}$, so $\frac{6}{4} + \frac{1}{4} = \frac{7}{4}$. Therefore, the answer is $x^{\frac{7}{4}}$, which is choice B. Looking at the wrong answers: Choice A ($x^{\frac{5}{4}}$) results from incorrectly adding $1 + \frac{1}{4}$ instead of $\frac{3}{2} + \frac{1}{4}$. Choice C ($x^{\frac{3}{2}}$) is just the first term, suggesting you forgot to include the second radical entirely. Choice D ($x^{\frac{13}{4}}$) might come from multiplying the exponents instead of adding them. Remember: when you see radicals being multiplied, immediately convert to fractional exponents. This transforms a potentially confusing radical problem into straightforward exponent arithmetic that follows familiar rules.