Expand: In (4a)3
The Correct Answer and Explanation is :
To expand ((4a)^3), we will use the property of exponents which states that ((xy)^n = x^n y^n). Here, (x) is (4), (y) is (a), and (n) is (3).
Step-by-Step Expansion
- Identify Components: We have (4) and (a) as the two components of the expression.
- Apply the Exponent: According to the exponent rule, we can rewrite the expression as:
[
(4a)^3 = (4)^3 (a)^3
] - Calculate Each Component:
- First, calculate (4^3):
[
4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64
] - Next, calculate (a^3):
[
a^3 = a \times a \times a
]
- Combine the Results: Now we can combine the results from both calculations:
[
(4a)^3 = 64a^3
]
Conclusion
The final expanded expression is:
[
(4a)^3 = 64a^3
]
Explanation
This expansion demonstrates the application of the distributive property of exponents. When raising a product to a power, we raise each factor in the product to that power. This is particularly useful in algebra when simplifying expressions or solving equations involving variables and constants.
Understanding how to manipulate and expand expressions with exponents is crucial in algebra. It allows for easier simplification and helps maintain clarity when working with polynomial expressions. Mastering these concepts prepares you for more complex algebraic tasks, such as factoring polynomials, solving equations, and understanding function behaviors in calculus.
Thus, ((4a)^3) exemplifies the principles of exponents, demonstrating how they can be applied systematically to break down more complex algebraic expressions into simpler forms.