The exponential form of $\sqrt{\sqrt{2} \times \sqrt{3}}$ is

### Concept & Formula To find the exponential form of nested surds, you must apply the product rule for radicals and then convert the radical notation into fractional exponents. $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ $$\sqrt[n]{x} = x^{\frac{1}{n}}$$ ### Step-by-Step Solution * **Given:** The expression is $\sqrt{\sqrt{2} \times \sqrt{3}}$. * **Calculation / Deduction:** 1. First, simplify the innermost part of the expression by multiplying the two square roots: $$\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$$ 2. Substitute this back into the original expression: $$\sqrt{\sqrt{6}}$$ 3. Convert the inner square root to a fractional exponent: $$\sqrt{6^{\frac{1}{2}}}$$ 4. Convert the outer square root to a fractional exponent: $$(6^{\frac{1}{2}})^{\frac{1}{2}}$$ 5. Apply the power of a power rule $(x^m)^n = x^{m \times n}$: $$6^{\frac{1}{2} \times \frac{1}{2}} = 6^{\frac{1}{4}}$$ ### Exam Strategy & Shortcut **Direct Transformation:** You can immediately rewrite nested square roots as a 4th root. Since $\sqrt{\sqrt{x}} = \sqrt[4]{x} = x^{\frac{1}{4}}$, you simply multiply the inner values ($2 \times 3 = 6$) and apply the $\frac{1}{4}$ power. You can solve this mentally in 2 seconds. ### Common Pitfall A very common mistake is adding the fractional exponents instead of multiplying them when dealing with nested roots, leading students to incorrectly choose $6^1$ or $6^{\frac{1}{2}}$. Always multiply powers when a power is raised to another power. ### Final Answer **Therefore, the correct answer is $6^{\frac{1}{4}}$.**