Question

Apply the Properties of Integer Exponents to generate equivalent expressions to

Answer 1

To apply the properties of integer exponents and generate equivalent expressions, we can use the following key properties:

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

Let's generate some equivalent expressions using these properties. We'll take an example expression and apply these properties to create equivalent forms.

**Example Expression:** \( x^3 \cdot x^5 \)

Using the **Product of Powers Property**:
\[
x^3 \cdot x^5 = x^{3+5} = x^8
\]

Now, let's apply the **Power of a Product Property** to another expression like \( (2x^2)^3 \):
\[
(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8 \cdot x^{2 \cdot 3} = 8x^6
\]

For an expression that involves division, consider \( \frac{a^5}{a^2} \):
Using the **Quotient of Powers Property**:
\[
\frac{a^5}{a^2} = a^{5-2} = a^3
\]

Let’s combine some of these properties. Given the expression \( (3xy^2)^2 \):
Using the **Power of a Product Property**:
\[
(3xy^2)^2 = 3^2 \cdot (x)^2 \cdot (y^2)^2 = 9 \cdot x^2 \cdot y^{2 \cdot 2} = 9x^2y^4
\]

If you have a specific expression in mind that you'd like to manipulate using these properties, please let me know, and I can provide a more tailored simplification!