Certainly! To apply the properties of integer exponents to generate an equivalent expression, let's review the relevant properties of integer exponents:
- Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
- Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\) (where \(a \neq 0\))
- Power of a Power: \((a^m)^n = a^{m \cdot n}\)
- Power of a Product: \((ab)^n = a^n b^n\)
- 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!