https://file.aiquickdraw.com/mathbot/file/1728117810848o3s1w55b.jpg

In the image, you provided two mathematical expressions:

1. A fraction involving exponents: 
   \[
   \frac{2^{7} \times 2^{3}}{81}
   \]

2. A polynomial expression involving multiplication and distribution:
   \[
   x^2(3 - 2x + x^2) = 3x^2 - 2x^3 + x^4
   \]

Let's break these down:

### 1. Simplifying the Exponential Fraction
The expression is \(\frac{2^7 \times 2^3}{81}\).

First, combine the exponents of \(2\) using the property \(a^m \times a^n = a^{m+n}\):

\[
2^7 \times 2^3 = 2^{7+3} = 2^{10}.
\]

Now, we have:

\[
\frac{2^{10}}{81}.
\]

Since \(2^{10} = 1024\), the expression becomes:

\[
\frac{1024}{81}.
\]

This is a simplified fraction that cannot be reduced further.

### 2. Polynomial Expansion
The expression is \(x^2(3 - 2x + x^2)\). To expand this:

\[
x^2 \times 3 = 3x^2,
\]
\[
x^2 \times (-2x) = -2x^3,
\]
\[
x^2 \times x^2 = x^4.
\]

Thus, the expanded form is:

\[
3x^2 - 2x^3 + x^4.
\]

Would you like more details or have any further questions?

---

Here are 5 related questions you might find interesting:

1. What are the general rules for simplifying exponents?
2. How can you further simplify a fraction involving large numbers?
3. Can you factor the polynomial \(3x^2 - 2x^3 + x^4\)?
4. What other properties of exponents are useful in simplifying expressions?
5. How do you graph the function \(3x^2 - 2x^3 + x^4\)?

**Tip:** When simplifying exponents, always look for opportunities to combine terms with the same base. This greatly simplifies calculations.

Simplify the fraction and expand the polynomial: \( \frac{2^7 \times 2^3}{81} \) and \( x^2(3 - 2x + x^2) \).

Learn how to simplify expressions involving exponents and expand polynomials using distributive property. Includes detailed steps for expressions like \( \frac{2^7 \times 2^3}{81} \) and \( x^2(3 - 2x + x^2) \). Suitable for Grades 9-11.

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