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 $a\ne 0$)
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 $b\ne 0$)

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 $(3x{y}^2{)}^2$: Using the Power of a Product Property: $$(3x{y}^2{)}^2=3^2\cdot (x{)}^2\cdot (y^2{)}^2=9\cdot x^2\cdot y^{2\cdot 2}=9x^2{y}^4$$

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!