Expand each expression using the Binomial Theorem....

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding any power of a binomial expression. It states that (a + b)^n can be expressed as a sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) represents the binomial coefficients. This theorem is widely used in algebra to simplify expressions, solve equations, and analyze polynomial behaviors.

Binomial Coefficients

Binomial coefficients, typically expressed as C(n, k) or (n choose k), are numerical factors that appear in the expansion of binomial expressions according to the Binomial Theorem. They are computed using factorials, with the formula C(n, k) = n! / (k!(n-k)!), and represent the number of ways to choose k elements from a set of n distinct elements.

General Term of a Binomial Expansion

The General Term in a binomial expansion is given by T(k+1) = C(n, k) * a^(n-k) * b^k, where k ranges from 0 to n. This term-by-term representation allows one to expand the expression by computing individual terms based on the values of a, b, n, and k, thereby making the process systematic and organized.

Handling Radicals in Exponents

When dealing with radical expressions in binomial expansions, careful treatment of the exponents is required. Radicals can be expressed as fractional exponents, and the standard rules of exponents apply. This conversion allows for seamless incorporation into the binomial formula, enabling correct calculation of each term's exponents in the expansion.

Transcript

00:01 We can use the binomial theorem to expand binomials.

00:05 If we have square root of x plus the square root of 2, all to the 6th, we can expand this using the binomial theorem.

00:15 First, we have to rewrite this so that it matches this form.

00:20 So we can rewrite this equation as x to the 1 half plus 2 to the 1 half to the 6th.

00:33 So we could see that in this expression, x equals x the one half a equals two to the one half and n equals six so for the first term j equals zero so we get six choose zero times x to the one half to the sixth times a to the zero which is just one the next term j equals one so we get six choose one times x to the one half all raised to the n minus j power.

01:20 So remember, n is still six and j equals one.

01:23 It's raised to the fifth power times our a, which is two to the one -half power to the first power, because j equals one.

01:37 For the third term, j equals two, so we get six choose two times x to the one -half to the fourth, times two to the one -half to the second.

01:50 Next, j equals 3, so we have 3, so we have 6 choose 3 times x to the 1 half to the 3 times a to the 1 half, sorry times 2 to the 1 half, also to the 3rd.

02:10 For the next term, j equals 4, so we get 6 shoes 4 times x to the 1 half to the 6 minus 4, so that's going to be squared, times 2 to the 1 half to the 4th.

02:27 The next term, j equals 5, so we get 6 to the 5th times x to the 1 half to the 6 minus 5, which equals 1, times 2 to the 1 half to the 1 half to the 5th.

02:47 Now for the last term, j equals 6.

02:49 So we get 6 to 6 to 6 times x to the 1 half to the 6 minus 6, which is x to the 1⁄2 to the 0 power, which equals 1 times 2 to the 1 half to the 6th.

03:04 So now all that's left is to simplify this equation.

03:10 Now to solve for, now to find x -chew -0 all the way to x -chew -6, we can do this by hand, or we can use the sixth row of pascal's triangle because this is a sixth degree equation.

03:28 So six to the zero equals one...

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