(x^7 * x^6)^5 / (x^4 * x^3)^3

Answer 1

To simplify the expression \((x^7 * x^6)^5 / (x^4 * x^3)^3\), we will follow these steps:

1. Apply the product of powers property, which states \(x^a * x^b = x^{a+b}\).
2. Use the power of a power property, which states \((x^a)^b = x^{a \cdot b}\).
3. Simplify the fraction using the quotient of powers property, which states \(\frac{x^a}{x^b} = x^{a-b}\).

Let's simplify the expression step-by-step:

1. Simplify the numerator: \[ (x^7 * x^6)^5 = (x^{7+6})^5 = (x^{13})^5 = x^{13 \cdot 5} = x^{65} \]

2. Simplify the denominator: \[ (x^4 * x^3)^3 = (x^{4+3})^3 = (x^{7})^3 = x^{7 \cdot 3} = x^{21} \]

3. Now we can write the complete expression: \[ \frac{x^{65}}{x^{21}} \]

4. Use the quotient of powers property: \[ x^{65 - 21} = x^{44} \]

Thus, the simplified expression is: \[ \boxed{x^{44}} \]