%5E3.jpeg "(4n-3)^3")
Exploring the Cube of (4n-3)
The expression (4n-3)³ represents the cube of the binomial (4n-3). This means we are multiplying the expression by itself three times:
(4n-3)³ = (4n-3) * (4n-3) * (4n-3)
To expand this, we can use the distributive property and some algebraic manipulation:
Expanding the Expression
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First, we multiply the first two factors:
(4n-3) * (4n-3) = 16n² - 12n - 12n + 9
= 16n² - 24n + 9 -
Now, we multiply the result by the remaining factor:
(16n² - 24n + 9) * (4n-3) = 64n³ - 48n² + 36n - 48n² + 36n - 27 = 64n³ - 96n² + 72n - 27
Therefore, the expanded form of (4n-3)³ is 64n³ - 96n² + 72n - 27.
Understanding the Pattern
Notice that the expanded form of the cube of a binomial always results in a polynomial with four terms:
- The first term: is the cube of the first term of the binomial (in this case, (4n)³ = 64n³).
- The second term: is three times the product of the square of the first term and the second term of the binomial ((3 * 4n² * -3) = -96n²).
- The third term: is three times the product of the first term and the square of the second term ((3 * 4n * (-3)²) = 72n).
- The fourth term: is the cube of the second term of the binomial ((-3)³ = -27).
This pattern applies to any binomial raised to the power of three.
Applications
Understanding the expansion of (4n-3)³ and similar expressions has applications in:
- Algebraic manipulation: Expanding and simplifying expressions involving binomials raised to a power.
- Calculus: Differentiating and integrating functions involving binomials.
- Physics: Solving equations involving variables raised to powers.
By understanding this basic expansion, we can tackle more complex problems involving binomials and their powers.