Question

Find the sum of the coefficients in the expansion of \(\left(5 x^{6}-\frac{4}{x^{9}}\right)^{10}\). (1) \(5^{10}\) (2) 1 (3) \(4^{10}\) (4) 0

Short answer

Answer: \(5^{10} + 5^8 + 5^6 + 5^4 + 5^2 - 1048576\)

Step-by-step solution

Write down the binomial expansion

Using the binomial theorem, we can write down the expansion of \(\left(5x^{6}-\frac{4}{x^{9}}\right)^{10}\) as: \[ \sum_{k=0}^{10} \binom{10}{k} \left(5x^{6}\right)^{10-k} \left(-\frac{4}{x^{9}}\right)^k \]

Simplify the expansion

Now, we can begin to simplify the expansion. Notice that when we multiply the terms inside the summation, we will end up with various powers of \(x\). Let's focus on finding the sum of coefficients of those powers of \(x\) for now. \[ \sum_{k=0}^{10} \binom{10}{k} \left(5^{10-k}\right) x^{60-6k}\left(-4^{k}\right) x^{-9k} \]

Combine like terms

Combine the powers of \(x\) in the expression above, we get \[ \sum_{k=0}^{10} \binom{10}{k} 5^{10-k}\left(-4\right)^k x^{60-15k} \]

Find the sum of the coefficients

In order to find the sum of the coefficients in the expansion, we need to sum the expression above. Since we are only interested in the coefficients, we can ignore the powers of \(x\). The sum becomes \[ \sum_{k=0}^{10} \binom{10}{k} 5^{10-k}\left(-4\right)^k \] Now, we can plug in the values of \(k\) from 0 to 10 and evaluate the expression: \[ \binom{10}{0} 5^{10}\left(-4\right)^0 + \binom{10}{1} 5^{9}\left(-4\right)^1 + \cdots + \binom{10}{10} 5^0 \left(-4\right)^{10} \] Evaluating the sum, we get \[ 1 \cdot 5^{10} + 10 \cdot 5^9(-4) + 45 \cdot 5^8(16) + 120 \cdot 5^7(-64) + 210 \cdot 5^6(256) + 252 \cdot 5^5(-1024) + 210 \cdot 5^4(4096) + 120 \cdot 5^3(-16384) + 45 \cdot 5^2(65536) + 10 \cdot 5^1(-262144) + 1 \cdot (-1048576) \] \[= 1 \cdot 5^{10} - 0 \cdot 5^9 + 1 \cdot 5^8 - 0 \cdot 5^7 + 1 \cdot 5^6 - 0 \cdot 5^5 + 1 \cdot 5^4 - 0 \cdot 5^3 + 1 \cdot 5^2 - 0 \cdot 5^1 + 1 \cdot (-1048576) \] The sum of the coefficients is \(5^{10} + 5^8 + 5^6 + 5^4 + 5^2 - 1048576\). By calculating the values of these expressions, we find the sum of the coefficients to be (1). \(5^{10}\).

Key concepts

Coefficients

In binomial expansion, coefficients are the numbers that multiply the variables in each term of the expansion. They play a crucial role in understanding the structure of the expanded expression. When expanding an expression like \((5x^{6}-\frac{4}{x^{9}})^{10}\), the coefficients are determined by the binomial coefficients, which can be calculated using the binomial formula. This involves combinations, often written as \(\binom{n}{k}\), where each \(k\) represents a specific term in the expansion. These coefficients can be found by using Pascal's triangle or directly calculating through combinations.

Binomial Theorem

The Binomial Theorem provides a powerful way to expand expressions raised to a power. It gives us a structured method to break down \((a+b)^n\) into a series of terms. This theorem uses binomial coefficients \(\binom{n}{k}\). Each term in the expansion is represented by \(\binom{n}{k} a^{n-k} b^k\).

For example, in the expansion of \((5x^{6}-\frac{4}{x^{9}})^{10}\), the theorem helps us understand how to systematically distribute powers between the terms and calculate each component of the expansion. Understanding the Binomial Theorem is key to not getting lost in the complexity of larger expansions.

Powers of Variables

In binomial expansions, variables take on different powers in each term. The power of a variable is determined by the position of the term within the expansion, as well as the original expression being expanded. For instance, in \((5x^{6}-\frac{4}{x^{9}})^{10}\), you’ll notice the powers of \(x\) change based on the exponents applied to \(5x^{6}\) and \(-\frac{4}{x^{9}}\).

By combining like terms, the net power of the variable can be calculated. Tracking this carefully is vital, especially when we're only interested in the coefficients, as we will ignore the powers of \(x\) when calculating their sum.

Sum of Coefficients

Finding the sum of the coefficients in a binomial expansion can sometimes simplify complex calculations. If you need the sum of the coefficients only, simply set the variables to 1. In the given problem, substituting \(x = 1\) in \((5x^{6}-\frac{4}{x^{9}})^{10}\) converts the problem into a numerical sum of coefficients without variables.

The expression then simplifies to evaluating the sum of terms given by \(\sum_{k=0}^{10} \binom{10}{k} 5^{10-k}(-4)^{k}\). This shows the importance of the binomial expression's terms when equating effectively to calculate the required sum.