Question

fully simplify (x^7 * x^6)^5
(x^4*x^3)^3

Answer 1

To simplify the expression \(\frac{(x^7 \cdot x^6)^5}{(x^4 \cdot x^3)^3}\), we will follow the rules of exponents.

**Step 1: Simplify the numerator**

Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can simplify the base in the numerator:

\[
x^7 \cdot x^6 = x^{7+6} = x^{13}
\]

Now we raise this to the power of 5:

\[
(x^{13})^5 = x^{13 \cdot 5} = x^{65}
\]

**Step 2: Simplify the denominator**

Similarly, we simplify the base in the denominator:

\[
x^4 \cdot x^3 = x^{4+3} = x^{7}
\]

Now we raise this to the power of 3:

\[
(x^{7})^3 = x^{7 \cdot 3} = x^{21}
\]

**Step 3: Divide the simplified numerator by the simplified denominator**

Now we have:

\[
\frac{x^{65}}{x^{21}}
\]

Using the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify this:

\[
x^{65 - 21} = x^{44}
\]

**Final Answer:**

The fully simplified expression is

\[
\boxed{x^{44}}
\]