Use the Law of Exponents to rewrite and simplify the expression √a√b/∛ab

To simplify the expression \(\frac{\sqrt{a} \cdot \sqrt{b}}{\sqrt[3]{ab}}\) using the Law of Exponents, follow these steps:

1. **Express square roots and cube roots as exponents:**

   - \(\sqrt{a} = a^{1/2}\)
   - \(\sqrt{b} = b^{1/2}\)
   - \(\sqrt[3]{ab} = (ab)^{1/3}\)

   Therefore, the expression becomes:

   \[
   \frac{a^{1/2} \cdot b^{1/2}}{(ab)^{1/3}}
   \]

2. **Rewrite the denominator:**

   - \((ab)^{1/3}\) can be expressed as \(a^{1/3} \cdot b^{1/3}\)

   So, the expression now is:

   \[
   \frac{a^{1/2} \cdot b^{1/2}}{a^{1/3} \cdot b^{1/3}}
   \]

3. **Apply the Law of Exponents (subtract exponents):**

   - For \(a\): \(\frac{a^{1/2}}{a^{1/3}} = a^{1/2 - 1/3} = a^{3/6 - 2/6} = a^{1/6}\)
   - For \(b\): \(\frac{b^{1/2}}{b^{1/3}} = b^{1/2 - 1/3} = b^{3/6 - 2/6} = b^{1/6}\)

   Thus, the simplified expression is:

   \[
   a^{1/6} \cdot b^{1/6}
   \]

4. **Combine the terms:**

   \[
   a^{1/6} \cdot b^{1/6} = (ab)^{1/6}
   \]

So, the simplified expression is \((ab)^{1/6}\).

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How would the expression change if we had \(\sqrt{a} \cdot \sqrt{b} \cdot \sqrt[3]{a}\) instead?
2. What if the denominator was \(\sqrt[3]{a^2b^2}\) instead of \(\sqrt[3]{ab}\)?
3. How do you simplify \(\frac{a^{2/3} \cdot b^{1/4}}{a^{1/2} \cdot b^{1/3}}\)?
4. What is the simplified form of \(\sqrt[4]{a^2b^3}\)?
5. How does the Law of Exponents apply when multiplying expressions with different bases?

**Tip:** When simplifying expressions involving roots and exponents, always convert the roots to fractional exponents before applying the laws of exponents.

Learn how to simplify the expression √a√b/∛ab using the Law of Exponents. This step-by-step solution covers key algebraic concepts including roots, fractional exponents, and exponent subtraction. Suitable for students in Grades 9-11.

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