apply the properties of integer exponents to generate an equivalent algebraic expression

Answer 1

Certainly! In order to generate an equivalent algebraic expression using the properties of integer exponents, let's start with an expression and then apply the properties. One of the foundational properties of exponents is:

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

Let’s consider the expression \( x^3 \cdot x^4 \).

Step 1: Apply the Product of Powers Property

Using the product of powers property:

\[ x^3 \cdot x^4 = x^{3+4} = x^7 \]

So, \( x^3 \cdot x^4 \) is equivalent to \( x^7 \).

Another Example

Let’s take a slightly more complex expression: \( \frac{y^5}{y^2} \).

Step 2: Apply the Quotient of Powers Property

Using the quotient of powers property:

\[ \frac{y^5}{y^2} = y^{5-2} = y^3 \]

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

One More Example

Consider \( (2x^3)^4 \).

Step 3: Apply the Power of a Product Property and Power of a Power Property

Using both properties:

\[ (2x^3)^4 = 2^4 \cdot (x^3)^4 = 16 \cdot x^{3 \cdot 4} = 16x^{12} \]

So, \( (2x^3)^4 \) is equivalent to \( 16x^{12} \).

If you have a specific expression you want to simplify or transform, please let me know!