To simplify $(8 \sqrt{a^3 b^6})^{\frac{1}{3}}$ using index laws, follow these steps:

---

Step 1: Rewrite the square root

Recall that $\sqrt{x} = x^{\frac{1}{2}}$. Thus: $\sqrt{a^3 b^6} = (a^3 b^6)^{\frac{1}{2}}$

Now substitute this back into the expression: $(8 \sqrt{a^3 b^6})^{\frac{1}{3}} = (8 (a^3 b^6)^{\frac{1}{2}})^{\frac{1}{3}}$

---

Step 2: Distribute the cube root

The entire expression is raised to the power of $\frac{1}{3}$, so apply the power to each part separately: $(8 (a^3 b^6)^{\frac{1}{2}})^{\frac{1}{3}} = 8^{\frac{1}{3}} \cdot \left((a^3 b^6)^{\frac{1}{2}}\right)^{\frac{1}{3}}$

---

Step 3: Simplify each part

Part 1: Simplify $8^{\frac{1}{3}}$

The cube root of 8 is 2, so: $8^{\frac{1}{3}} = 2$

Part 2: Simplify the powers of $a^3 b^6$

When raising a power to another power, multiply the exponents: $\left((a^3 b^6)^{\frac{1}{2}}\right)^{\frac{1}{3}} = (a^3 b^6)^{\frac{1}{2} \cdot \frac{1}{3}} = (a^3 b^6)^{\frac{1}{6}}$

Now distribute $\frac{1}{6}$ to both $a^3$ and $b^6$: $(a^3 b^6)^{\frac{1}{6}} = a^{\frac{3}{6}} b^{\frac{6}{6}} = a^{\frac{1}{2}} b^1 = \sqrt{a} b$

---

Step 4: Combine the results

Now combine everything: $(8 \sqrt{a^3 b^6})^{\frac{1}{3}} = 2 \cdot (\sqrt{a} b) = 2b\sqrt{a}$

---

Final Answer:

$(8 \sqrt{a^3 b^6})^{\frac{1}{3}} = 2b\sqrt{a}$

---

Would you like further clarification or details on any step? 😊

---

Related Questions:

1. How are fractional exponents related to roots, and how can we simplify them?
2. What is the process for simplifying cube roots of larger numbers?
3. How can index laws be applied to simplify expressions with multiple variables?
4. Why does the square root simplify to $\frac{1}{2}$ as an exponent?
5. How can this method be used in real-world applications involving roots?

Tip:

When simplifying expressions involving roots and powers, always handle constants and variables separately to keep your work clear and organized.