1. $$ \frac{\sqrt[4]{\left(a^{2} b^{2} ight)^{6}}}{a^{5} b^{7} imes a^{-2} b^{-4}} $$

Question

1. $$ \frac{\sqrt[4]{\left(a^{2} b^{2}ight)^{6}}}{a^{5} b^{7} 	imes a^{-2} b^{-4}} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\sqrt[4]{\left(a^{2}b^{2}\right)^{6}}}{a^{5}b^{7}a^{-2}b^{-4}}$$
- step1: Simplify the root:
  $$\frac{\sqrt{\left(a^{2}b^{2}\right)^{3}}}{a^{5}b^{7}a^{-2}b^{-4}}$$
- step2: Simplify the root:
  $$\frac{a^{2}b^{2}\sqrt{a^{2}b^{2}}}{a^{5}b^{7}a^{-2}b^{-4}}$$
- step3: Multiply:
  $$\frac{a^{2}b^{2}\sqrt{a^{2}b^{2}}}{a^{3}b^{3}}$$
- step4: Reduce the fraction:
  $$\frac{\sqrt{a^{2}b^{2}}}{ab}$$
Calculate or simplify the expression $$ \frac{\sqrt{a^{2}b^{2}}}{ab} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\sqrt{a^{2}b^{2}}}{ab}$$
- step1: Simplify the root:
  $$\frac{ab}{ab}$$
- step2: Divide the terms:
  $$1$$
Let's simplify the expression step by step:

1. **Original Expression:**
   $$
   \frac{\sqrt[4]{\left(a^{2} b^{2}\right)^{6}}}{a^{5} b^{7} \times a^{-2} b^{-4}}
   $$

2. **Simplifying the Numerator:**
   The numerator is $$\sqrt[4]{(a^{2} b^{2})^{6}}$$. We can simplify this as follows:
   $$
   (a^{2} b^{2})^{6} = a^{12} b^{12}
   $$
   Therefore,
   $$
   \sqrt[4]{a^{12} b^{12}} = (a^{12} b^{12})^{\frac{1}{4}} = a^{3} b^{3}
   $$

3. **Simplifying the Denominator:**
   The denominator is $$a^{5} b^{7} \times a^{-2} b^{-4}$$. We can combine the terms:
   $$
   a^{5} \times a^{-2} = a^{5 - 2} = a^{3}
   $$
   $$
   b^{7} \times b^{-4} = b^{7 - 4} = b^{3}
   $$
   Thus, the denominator simplifies to:
   $$
   a^{3} b^{3}
   $$

4. **Putting it Together:**
   Now we can rewrite the entire expression:
   $$
   \frac{a^{3} b^{3}}{a^{3} b^{3}} = 1
   $$

Therefore, the final simplified result is:
$$
\boxed{1}
$$
The expression simplifies to 1.

1. \( \frac{\sqrt[4]{\left(a^{2} b^{2}\right)^{6}}}{a^{5} b^{7} \times a^{-2} b^{-4}} \)

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