b) $$ \left(\frac{1}{9} m^{5}\right)^{2}\left(\frac{1}{4} m^{7}\right)^{3} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
\left(\frac{1}{9} m^{5}\right)^{2}\left(\frac{1}{4} m^{7}\right)^{3}
$$

**Step 1: Apply the Power to Each Factor**

First, apply the exponent to each part inside the parentheses:

$$
\left(\frac{1}{9}\right)^{2} \cdot (m^{5})^{2} \cdot \left(\frac{1}{4}\right)^{3} \cdot (m^{7})^{3}
$$

**Step 2: Simplify Each Part**

Calculate each part separately:

$$
\left(\frac{1}{9}\right)^{2} = \frac{1}{81}
$$
$$
(m^{5})^{2} = m^{10}
$$
$$
\left(\frac{1}{4}\right)^{3} = \frac{1}{64}
$$
$$
(m^{7})^{3} = m^{21}
$$

**Step 3: Multiply the Simplified Parts Together**

Now, multiply all the simplified parts:

$$
\frac{1}{81} \cdot m^{10} \cdot \frac{1}{64} \cdot m^{21} = \frac{1}{81 \times 64} \cdot m^{10+21}
$$

**Step 4: Final Simplification**

Calculate the numerical part and combine the exponents:

$$
81 \times 64 = 5184
$$
$$
m^{10+21} = m^{31}
$$

So, the simplified expression is:

$$
\frac{m^{31}}{5184}
$$

**Final Answer:**

$$
\boxed{\dfrac{m^{31}}{5184}}
$$
$$ \frac{m^{31}}{5184} $$

b) \( \left(\frac{1}{9} m^{5}\right)^{2}\left(\frac{1}{4} m^{7}\right)^{3} \)

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