To apply the properties of integer exponents and generate equivalent expressions, we can use the following key properties:
- Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \)
- Quotient of Powers Property: \( \frac{a^m}{a^n} = a^{m-n} \) (where \( a \neq 0 \))
- Power of a Power Property: \( (a^m)^n = a^{m \cdot n} \)
- Power of a Product Property: \( (ab)^n = a^n \cdot b^n \)
- 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!
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