Question

question:
which expression is equivalent to ((64r^{4}s^{12})^{\frac{1}{2}})?
option 1: (8r^{4}s^{12})
option 2: (32r^{4}s^{12})
option 3: (8r^{2}s^{6})
option 4: (64r^{2}s^{6})

Explanation:

Step1: Apply power rule to each term

Use the exponent rule $(ab)^n=a^n b^n$ and $(a^m)^n=a^{m \cdot n}$:
$$(64r^4s^{12})^{\frac{1}{2}} = 64^{\frac{1}{2}} \cdot (r^4)^{\frac{1}{2}} \cdot (s^{12})^{\frac{1}{2}}$$

Step2: Calculate each component

Compute each term separately:
$64^{\frac{1}{2}} = \sqrt{64} = 8$
$(r^4)^{\frac{1}{2}} = r^{4 \cdot \frac{1}{2}} = r^2$
$(s^{12})^{\frac{1}{2}} = s^{12 \cdot \frac{1}{2}} = s^6$

Step3: Combine the results

Multiply the simplified terms together:
$$8 \cdot r^2 \cdot s^6 = 8r^2s^6$$

Answer:

Option 3. $8r^2s^6$