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

Options

A. `b ^(3) x ^(7)a ^(3) z ^(3)`
B. `b ^(2) x ^(7) a ^(3) z ^(3)`
C. `b ^(2) x^(5) a ^(3) z ^(3)`
D. `b ^(2) x ^(7)a ^(3) z ^(2)`

Answer

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

### Step 1: Combine the Numerator
First, we will multiply the two expressions in the numerator:

\[ (b^{2} x^{3} a^{2} z^{4}) \cdot (b^{2} x^{4} a^{4} z^{3}) = b^{2} \cdot b^{2} \cdot x^{3} \cdot x^{4} \cdot a^{2} \cdot a^{4} \cdot z^{4} \cdot z^{3} \]

Using the property of exponents \(a^m \cdot a^n = a^{m+n}\), we get:

\[ = b^{2+2} \cdot x^{3+4} \cdot a^{2+4} \cdot z^{4+3} = b^{4} \cdot x^{7} \cdot a^{6} \cdot z^{7} \]

### Step 2: Write the Complete Expression
Now we can write the complete expression:

\[ \frac{b^{4} x^{7} a^{6} z^{7}}{a^{3} b^{2} z^{4}} \]

### Step 3: Simplify the Denominator
Now we will simplify the expression by dividing each term in the numerator by the corresponding term in the denominator:

\[ = \frac{b^{4}}{b^{2}} \cdot \frac{x^{7}}{1} \cdot \frac{a^{6}}{a^{3}} \cdot \frac{z^{7}}{z^{4}} \]

Using the property of exponents again:

\[ = b^{4-2} \cdot x^{7} \cdot a^{6-3} \cdot z^{7-4} = b^{2} \cdot x^{7} \cdot a^{3} \cdot z^{3} \]

### Final Answer
Thus, the simplified expression is:

\[ b^{2} x^{7} a^{3} z^{3} \]