(a^2b^3)^4

2 min read Jun 16, 2024
(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.

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