Question
2. \( (2 a+b)^{3}-1 \)
Ask by Mitchell Hilton. in South Africa
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \( 8a^{3}+12a^{2}b+6ab^{2}+b^{3}-1 \).
Solution
Calculate or simplify the expression \( (2a+b)^3-1 \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(2a+b\right)^{3}-1\)
- step1: Expand the expression:
\(8a^{3}+12a^{2}b+6ab^{2}+b^{3}-1\)
The simplified form of the expression \( (2a+b)^3-1 \) is \( 8a^{3}+12a^{2}b+6ab^{2}+b^{3}-1 \).
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Simplify this solution The Deep Dive
To expand \( (2a + b)^3 - 1 \), we first use the binomial theorem, which tells us how to expand binomials raised to powers. The expansion gives us: \[ (2a + b)^3 = (2a)^3 + 3(2a)^2(b) + 3(2a)(b)^2 + (b)^3 \] Calculating each term, we find: - \( (2a)^3 = 8a^3 \) - \( 3(2a)^2(b) = 3 \cdot 4a^2b = 12a^2b \) - \( 3(2a)(b)^2 = 6ab^2 \) - \( (b)^3 = b^3 \) Putting it all together, we have: \[ (2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3 \] Subtracting 1 gives the final result: \[ (2a + b)^3 - 1 = 8a^3 + 12a^2b + 6ab^2 + b^3 - 1 \]
