Question

apply the properties of integer exponents to generate an equivalent

Answer 1

Certainly! To apply the properties of integer exponents to generate an equivalent expression, let's review the relevant properties of integer exponents:

1. **Product of Powers**: \(a^m \cdot a^n = a^{m+n}\)
2. **Quotient of Powers**: \(\frac{a^m}{a^n} = a^{m-n}\) (where \(a \neq 0\))
3. **Power of a Power**: \((a^m)^n = a^{m \cdot n}\)
4. **Power of a Product**: \((ab)^n = a^n b^n\)
5. **Power of a Quotient**: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) (where \(b \neq 0\))

Let's use these properties to generate an equivalent expression.

**Example Expression**:
Let's consider the expression
\[
x^3 \cdot x^5
\]

**Step 1**: Apply the Product of Powers property.
Using the Product of Powers property:
\[
x^3 \cdot x^5 = x^{3+5} = x^8
\]

So, \(x^3 \cdot x^5\) is equivalent to \(x^8\).

**Another Example**:
Now let's consider a more complex expression:
\[
\frac{2^5}{2^2}
\]

**Step 2**: Apply the Quotient of Powers property.
Using the Quotient of Powers property:
\[
\frac{2^5}{2^2} = 2^{5-2} = 2^3
\]

Thus, \(\frac{2^5}{2^2}\) is equivalent to \(2^3\).

If you have a specific expression in mind that you'd like to simplify using the properties of integer exponents, please provide it, and I can help with that!