https://file.aiquickdraw.com/mathbot/file/1725745488084ypqhvi9a.jpg

I see that you've uploaded an image of a math problem. The text for **Question 8 (part one)** says:

**"Use the Binomial Theorem to expand each of the following: (i) \( (1 + 2x)^6 \)"**

Let’s solve it step by step using the **Binomial Theorem**, which is given by the formula:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here:
- \( a = 1 \)
- \( b = 2x \)
- \( n = 6 \)

Now let's apply the expansion:

\[
(1 + 2x)^6 = \sum_{k=0}^{6} \binom{6}{k} (1)^{6-k} (2x)^k
\]

Simplifying further:
\[
(1 + 2x)^6 = \sum_{k=0}^{6} \binom{6}{k} (2x)^k
\]

Now expand each term:

\[
= \binom{6}{0}(2x)^0 + \binom{6}{1}(2x)^1 + \binom{6}{2}(2x)^2 + \binom{6}{3}(2x)^3 + \binom{6}{4}(2x)^4 + \binom{6}{5}(2x)^5 + \binom{6}{6}(2x)^6
\]

Calculate the binomial coefficients and powers of \(2x\):

\[
= 1(1) + 6(2x) + 15(4x^2) + 20(8x^3) + 15(16x^4) + 6(32x^5) + 1(64x^6)
\]

Simplify each term:

\[
= 1 + 12x + 60x^2 + 160x^3 + 240x^4 + 192x^5 + 64x^6
\]

Thus, the expansion of \( (1 + 2x)^6 \) is:

\[
1 + 12x + 60x^2 + 160x^3 + 240x^4 + 192x^5 + 64x^6
\]

Would you like further details on the steps, or do you have any other questions?

Here are five related questions:
1. How does the Binomial Theorem work for negative exponents?
2. Can you use the Binomial Theorem for fractional powers?
3. What happens if \( x = 0 \) in the expanded binomial expression?
4. How does Pascal's Triangle relate to the binomial expansion?
5. How do you calculate the binomial coefficients \( \binom{n}{k} \)?

**Tip**: When expanding binomials, carefully compute each binomial coefficient and power of terms to avoid errors.

Use the Binomial Theorem to expand (1 + 2x)^6.

Expand (1 + 2x)^6 using the Binomial Theorem. This step-by-step solution covers key algebraic concepts such as binomial coefficients and powers. Perfect for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation