Math Problem Statement
Solution
The image contains the following mathematical problem:
Part (i): Find the binomial expansion of up to and including the term in , simplifying the coefficients.
Part (ii): Given that , use your binomial expansion in part (i) to obtain the value of , giving the answer to five significant places.
Part (i): Binomial Expansion
The expression given is .
To simplify the binomial expansion, we can factor out the 27:
Now, we use the binomial expansion formula for where and :
Expanding :
This simplifies to:
Thus, multiplying by 3 (from the earlier factoring out of 27):
Including up to the term, we have:
Part (ii): Using Binomial Expansion to Estimate
We are given .
We know that .
To estimate , we can use the binomial expansion we found in part (i). By setting , we substitute this into the expansion from part (i):
Therefore,
[ \sqrt[3]{3} = \frac{3}{\sqrt[3]{9}} \approx \frac{3}{3.1975} = 0.9383 ]
Finally, rounding to five significant places:
Would you like more details on any step?
Here are some related questions:
- How do we derive the general binomial expansion for fractional powers?
- Why is binomial expansion useful for approximations like these?
- What happens if we include more terms in the binomial expansion?
- How can we refine the approximation further using calculus methods?
- Can we use this method to approximate other roots, such as ?
Tip: When using binomial expansions, remember to watch for how many terms you need to include for the desired accuracy.
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Math Problem Analysis
Mathematical Concepts
Binomial Expansion
Algebra
Approximations
Formulas
Binomial expansion formula: (1 + u)^n = 1 + n*u + (n(n-1)/2!) * u^2 + (n(n-1)(n-2)/3!) * u^3 + ...
Cube roots approximation
Theorems
Binomial Theorem
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Grades 11-12)