Expand each expression using the Binomial Theorem....

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. It expresses the expansion as a sum of terms, each involving a binomial coefficient, a power of a, and a power of b. This theorem is widely used in algebra and combinatorics to simplify and compute power expansions.

Binomial Coefficients

Binomial coefficients, represented as C(n, k) or 'n choose k', give the weights for each term in the expansion. They are calculated using factorials and can also be found in Pascal's Triangle. These coefficients are fundamental in determining how the terms of the expansion combine to form the final expression.

Handling Radicals in Exponents

When a binomial expression includes radical terms, these radicals are typically expressed as fractional exponents. The properties of exponents are then applied in the expansion process, ensuring that each term is correctly computed even when the bases involve radicals. This approach allows one to systematically expand and simplify the expression involving square roots or other radical forms.

Transcript

00:01 If we have the binomial, the square root of x minus the square root of 3, all to the 4th, we can use the binomial theorem to expand this.

00:11 But first, we want it in this form.

00:13 So you can rewrite this equation as x to the 1 half plus negative 3 to the 1 half, all to the 4th.

00:27 So we could see that the x here is equal to x to the 1 half.

00:33 The a here is equal to negative 3 to the 1 half, and the n over here is equal to 4.

00:49 So if we read this out, for the first term, j equals 0.

00:54 So we have n, which equals 4, choose j, which equals 0, times x to the n minus j.

01:03 So remember, in this case, rx, is x to the 1 half.

01:08 So we have x to the 1 half to the power of n minus j, n equals 4, j equals 0, so that's x to the 1 half, all to the 4th, times a, which in this case is negative 3 to the 1 half to the j.

01:26 In this case, j equals 0, so that's all to the 0 power.

01:30 Next term, j equals 1.

01:32 So you get 4, choose 1, times x to the 1 half, the 3 because 4 minus 1 equals 3.

01:43 Times negative 3 to the 1 half to the first power.

01:49 The next term, j equals 2.

01:50 So you get 4 choose 2 times x to the 1 half raised the power of n minus j.

01:59 N is still 4.

02:00 J equals 2.

02:01 So that's x to the 1 1 1 half squared times negative 3 to the 1 half to the j.

02:07 So negative 3 to the 1 half.

02:12 Next term, j equals 3.

02:14 So you get 4 choose 3 times x to the 1 half to the 1st times negative 3.

02:23 Negative 3 to the 1 half to the third.

02:29 Now the last term, j equals 4.

02:32 So you get 4 choose 4 times x to the 1 half to the 4 minus 4, which is 0 times negative 3 to the 1 half to the 4th.

02:46 Now the last step is just to simplify this.

02:50 To find these coefficients, you can do it by hand or you could use pascal's triangle...

Please give Ace some feedback