Chapter 12: Q. 12.70. (page 1209)

Use the Binomial Theorem to expand ${(m + n)}^{6}$ .

Short Answer

Expert verified

The expansion of the given binomial is ${(m + n)}^{6} = m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{3} n^{4} + 6 m^{2} n^{5} + n^{6}$

Step by step solution

Step 1. Given information.

The given expression is ${(m + n)}^{6}$ .

Step 2. Substitute the values and simplify.

Substitute in the values into the binomial expansion,

$a = m b = n n = 6$

Use the Binomial Theorem,

${(a + b)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) a^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a^{n - 1} b + (\begin{matrix} n \\ 2 \end{matrix}) a^{n - 2} b^{2} + . . . . . . . + (\begin{matrix} n \\ r \end{matrix}) a^{n - r} b^{r} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) b^{n}$

Let's put the values into the formula.

${(m + n)}^{6} = (\begin{matrix} 6 \\ 0 \end{matrix}) m^{6} + (\begin{matrix} 6 \\ 1 \end{matrix}) m^{6 - 1} n + (\begin{matrix} 6 \\ 2 \end{matrix}) m^{6 - 2} n^{2} + (\begin{matrix} 6 \\ 3 \end{matrix}) m^{6 - 3} n^{3} + (\begin{matrix} 6 \\ 4 \end{matrix}) m^{6 - 4} n^{4} + (\begin{matrix} 6 \\ 5 \end{matrix}) m^{6 - 5} n^{5} + (\begin{matrix} 6 \\ 6 \end{matrix}) n^{6}$

Simplify the exponents.

${(m + n)}^{6} = (\begin{matrix} 6 \\ 0 \end{matrix}) m^{6} + (\begin{matrix} 6 \\ 1 \end{matrix}) m^{5} n + (\begin{matrix} 6 \\ 2 \end{matrix}) m^{4} n^{2} + (\begin{matrix} 6 \\ 3 \end{matrix}) m^{3} n^{3} + (\begin{matrix} 6 \\ 4 \end{matrix}) m^{2} n^{4} + (\begin{matrix} 6 \\ 5 \end{matrix}) m n^{5} + (\begin{matrix} 6 \\ 6 \end{matrix}) n^{6}$

Step 3. Simplify the coefficients and exponents.

Evaluate the coefficients. Remember,

$(\begin{matrix} n \\ 1 \end{matrix}) = n, (\begin{matrix} n \\ n \end{matrix}) = 1, (\begin{matrix} n \\ 0 \end{matrix}) = 1$

${(m + n)}^{6} = 1 m^{6} + 6 m^{5} n + \frac{6!}{2! (4)!} m^{4} n^{2} + \frac{6!}{3! (3)!} m^{3} n^{3} + \frac{6!}{4! (2)!} m^{2} n^{4} + \frac{6!}{5! (1!)} m n^{5} + n^{6}$

Step 4. Conclusion.

Therefore, the expansion of the given expression is

${(m + n)}^{6} = m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + n^{6}$

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Use the Binomial Theorem to expand ${(m + n)}^{6}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Simplify the coefficients and exponents.

Step 4. Conclusion.

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Use the Binomial Theorem to expandm+n6.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Simplify the coefficients and exponents.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Use the Binomial Theorem to expand ${(m + n)}^{6}$ .