Binomial Theorem: Find Expansion & Approximation of 97^(1/2)

In summary, the conversation discusses finding the first four terms in the expansion of a given expression and using it to approximate the square root of 97. The first two terms are determined to be correct, but the subsequent terms are incorrect and need to be recalculated. The estimated value for the square root of 97 is also slightly off and the method used to solve it is questioned.
  • #1
naden1
1
0

Homework Statement



Find the first four terms in the expansion of [itex]\left(1-3x\right)^{3/2}[/itex]. By substituting in a suitable value of x, find an approximation to [itex]97^{1/2}[/itex].

Homework Equations



The Attempt at a Solution



I used the binomial expansion formula to work the answer and it is 1- 4.5x - (22x2/8) - (247x3/48) + ... .

Is that correct?

I did the second part and i got 0.848. What do you think?

Thanks in advance for any help.
 
Physics news on Phys.org
  • #2
naden1 said:
I used the binomial expansion formula to work the answer and it is 1- 4.5x - (22x2/8) - (247x3/48) + ... .

First two terms are correct, it goes wrong after that. Check your calculations again.
 
  • #3
naden1 said:
I did the second part and i got 0.848. What do you think?

I think that it's slightly off :wink:
 
Last edited:
  • #4
naden1 said:
I did the second part and i got 0.848. What do you think?

The positive square root of 100 is 10, and square root 81 is 9. Where would you expect square root of 97 to lie in? :wink:

How did you solve this part?
 
  • #5
Mentallic said:
I think that it's a slightly off :wink:

Slightly? Wouldn't that be too much? :-p
 

Related to Binomial Theorem: Find Expansion & Approximation of 97^(1/2)

What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula that allows us to expand binomials, which are expressions with two terms, to any power. It is written as (a + b)^n = a^n + na^(n-1)b + (n(n-1)/2!)a^(n-2)b^2 + ... + b^n, where n is a positive integer.

How do you find the expansion of a binomial using the Binomial Theorem?

To find the expansion of a binomial using the Binomial Theorem, we first identify the values of a, b, and n. Then, we use the formula (a + b)^n = a^n + na^(n-1)b + (n(n-1)/2!)a^(n-2)b^2 + ... + b^n to calculate each term and combine them to get the expanded form.

What is the significance of the Binomial Theorem in mathematics?

The Binomial Theorem has many applications in mathematics, especially in algebra and calculus. It allows us to simplify and solve complex equations, and also helps in understanding the patterns and relationships between numbers.

How can the Binomial Theorem be used to approximate a number raised to a fractional power?

The Binomial Theorem can be used to approximate a number raised to a fractional power by taking a term from the binomial expansion that has the same exponent as the fractional power. For example, to approximate 97^(1/2), we can use the term with n = 1/2 in the expansion of (9 + 7)^n.

Can the Binomial Theorem be used for any type of number and power?

Yes, the Binomial Theorem can be used for any type of number and power, as long as the values of a, b, and n are known. It is a general formula that can be applied to both positive and negative numbers, and any integer or fractional power.

Similar threads

Binomial Theorem - determine the term with...
Replies
18
Views
1K
Find the ##r^{th}## term of ##(a+2x)^n##
Replies
3
Views
794
Binomial Expansion: Evaluating Coefficient from two binomials
Replies
7
Views
2K
Use of binomial theorem in a sum of binomial coefficients?
Replies
1
Views
2K
Need help understanding the binomial series
Replies
3
Views
2K
What are the constants in the binomial series expansion for (1+mx)^-n?
Replies
34
Views
4K
Solve Binomial Theorem Homework: Find Coefficients of Degree 17 & x7
Replies
15
Views
2K
Range of validity for Binomial Series
Replies
2
Views
2K
Where Did ##(n−r+1)^{th}## Come From in Binomial Expansion?
Replies
3
Views
851
How do I expand (1 + x)^{2}(1 - 5x)^{14} as a series of powers of x?
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top