%5E4.jpeg "(a^2b^3)^4")
Simplifying (a²b³)^4
In mathematics, we often encounter expressions with exponents and variables. Understanding how to simplify these expressions is crucial. Let's take a look at the expression (a²b³)^4.
Applying the Power of a Product Rule
The expression involves raising a product of powers to another power. To simplify this, we use the power of a product rule:
(xy)^n = x^n * y^n
Applying this rule to our expression:
(a²b³)^4 = (a²)⁴ * (b³)⁴
Applying the Power of a Power Rule
Now, we need to simplify further by raising each term within the parentheses to the power of 4. We use the power of a power rule:
(x^m)^n = x^(m*n)
Applying this rule:
(a²)⁴ * (b³)⁴ = a^(24) * b^(34)
Simplifying the Expression
Finally, we perform the multiplications in the exponents:
a^(24) * b^(34) = a⁸ * b¹²
Therefore, the simplified form of (a²b³)^4 is a⁸b¹².
Key Takeaways
- The power of a product rule allows us to distribute the exponent to each factor within the parentheses.
- The power of a power rule simplifies the expression by multiplying the exponents.
By understanding and applying these rules, we can effectively simplify complex expressions involving exponents and variables.