%5E4.jpeg "(-3ab^3)^4")
Simplifying Expressions with Exponents: (-3ab^3)^4
In mathematics, simplifying expressions with exponents often involves applying the rules of exponents. Let's break down how to simplify the expression (-3ab^3)^4.
Understanding the Rules
The key to simplifying this expression lies in understanding the following rules of exponents:
- Product of Powers: (a^m)^n = a^(m*n)
- Power of a Product: (ab)^n = a^n * b^n
- Power of a Quotient: (a/b)^n = a^n / b^n
- Negative Exponent: a^-n = 1/a^n
Applying the Rules
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Applying the Power of a Product Rule: We begin by applying the Power of a Product Rule to the expression:
(-3ab^3)^4 = (-3)^4 * a^4 * (b^3)^4
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Applying the Product of Powers Rule: Next, we apply the Product of Powers Rule to simplify (b^3)^4:
(-3)^4 * a^4 * (b^3)^4 = (-3)^4 * a^4 * b^(3*4)
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Simplifying the Expression: Finally, we calculate the numerical exponent and simplify:
(-3)^4 * a^4 * b^(3*4) = 81a^4b^12
Conclusion
By applying the rules of exponents, we have successfully simplified the expression (-3ab^3)^4 to 81a^4b^12. Remember to always follow the order of operations (PEMDAS) when simplifying expressions.