9. $$ \frac{9 a^{7} b\left(3 a^{6} b^{2}\right)}{18\left(a^{5} b^{3}\right)} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{9 a^{7} b \cdot \left(3 a^{6} b^{2}\right)}{18 \cdot \left(a^{5} b^{3}\right)}
$$

**Step 1: Multiply the Numerator**

Multiply the constants and combine the like terms for $$ a $$ and $$ b $$:

$$
9 \cdot 3 = 27
$$
$$
a^{7} \cdot a^{6} = a^{13}
$$
$$
b \cdot b^{2} = b^{3}
$$

So, the numerator becomes:

$$
27 a^{13} b^{3}
$$

**Step 2: Multiply the Denominator**

Multiply the constants and retain the variables:

$$
18 \cdot a^{5} b^{3} = 18 a^{5} b^{3}
$$

**Step 3: Form the Fraction**

$$
\frac{27 a^{13} b^{3}}{18 a^{5} b^{3}}
$$

**Step 4: Simplify the Coefficients**

$$
\frac{27}{18} = \frac{3}{2}
$$

**Step 5: Subtract the Exponents for Like Bases**

$$
a^{13} / a^{5} = a^{8}
$$
$$
b^{3} / b^{3} = b^{0} = 1
$$

**Final Simplified Expression:**

$$
\frac{3}{2} a^{8}
$$

**Alternatively, it can be written as:**

$$
\frac{3a^{8}}{2}
$$
The simplified expression is $$ \frac{3a^{8}}{2} $$.

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