Simplify `(b ^(5) x ^(2) a ^(3) z ^(4)) ^(**) (b ^(3) x ^(2) a ^(4) z ^(5))//(a ^(2) b ^(3) z ^(2))`

Options

A. `b ^(5) x ^(4) a ^(5) z^(5)`
B. `b ^(5) x ^(4) a ^(5) z^(7)`
C. `b ^(5) x ^(4) a ^(4) z^(7)`
D. `b ^(4) x^(4) a ^(5) z^(7)`

Answer

To simplify the expression \((b^{5} x^{2} a^{3} z^{4}) \cdot (b^{3} x^{2} a^{4} z^{5}) / (a^{2} b^{3} z^{2})\), we can follow these steps:

### Step 1: Combine the Numerator
First, we need to multiply the two expressions in the numerator:
\[ (b^{5} x^{2} a^{3} z^{4}) \cdot (b^{3} x^{2} a^{4} z^{5}) = b^{5} \cdot b^{3} \cdot x^{2} \cdot x^{2} \cdot a^{3} \cdot a^{4} \cdot z^{4} \cdot z^{5} \]

### Step 2: Apply the Laws of Exponents
Using the laws of exponents, we can combine the powers:
\[ b^{5+3} \cdot x^{2+2} \cdot a^{3+4} \cdot z^{4+5} = b^{8} \cdot x^{4} \cdot a^{7} \cdot z^{9} \]

### Step 3: Write the Complete Expression
Now, we can write the complete expression:
\[ \frac{b^{8} x^{4} a^{7} z^{9}}{a^{2} b^{3} z^{2}} \]

### Step 4: Simplify the Expression
Now we will simplify the expression by dividing the terms:
- For \(b\):
\[ b^{8} / b^{3} = b^{8-3} = b^{5} \]

- For \(x\):
\[ x^{4} \text{ (no division, remains as is)} \]

- For \(a\):
\[ a^{7} / a^{2} = a^{7-2} = a^{5} \]

- For \(z\):
\[ z^{9} / z^{2} = z^{9-2} = z^{7} \]

### Step 5: Combine the Results
Putting it all together, we get:
\[ b^{5} x^{4} a^{5} z^{7} \]

### Final Answer
Thus, the simplified expression is:
\[ b^{5} x^{4} a^{5} z^{7} \]

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