Problem 28 Expand the binomial. $$\left(2... [FREE SOLUTION]

Chapter 6: Problem 28

Expand the binomial. $$\left(2 r^{-1}+s^{-1}\right)^{5}$$

Short Answer

Expert verified

The binomial expansion of $(2 r^{-1} + s^{-1})^5$ is \[ 32 r^{-5} + 80 r^{-4} s^{-1} + 80 r^{-3} s^{-2} + 40 r^{-2} s^{-3} + 10 r^{-1} s^{-4} + s^{-5} \]

Step by step solution

Identify the Binomial Coefficients

We first identify the coefficients of our binomial by using the formula for combinations on each term. In a binomial expansion $(a+b)^n$, the coefficients for $a$ and $b$ for each term in the expansion are determined by $n C k = n! / [(n-k)! * k!]$ where $n$ is the power to which the binomial is raised and $k$ is the index of term starting from 0. For the binomial in question $(2 r^{-1} + s^{-1})^5$, the coefficients will be: $5 C 0, 5 C 1, 5 C 2, 5 C 3, 5 C 4, 5 C 5$.

Calculate the Binomial Coefficients

Now, compute the coefficients: $5 C 0 = 1$, $5 C 1 = 5$, $5 C 2 = 10$, $5 C 3 = 10$, $5 C 4 = 5$, and $5 C 5 = 1$.

Apply the Binomial Theorem

Now, we apply the binomial theorem to expand the given expression, which is $ (2 * r^{-1} + s^{-1})^5 $. Following the binomial theorem, the expansion will be: \[1*(2 * r^{-1})^5 * (s^{-1})^0 + 5*(2 * r^{-1})^4 * (s^{-1})^1 + 10*(2 * r^{-1})^3 * (s^{-1})^2 + 10*(2 * r^{-1})^2 * (s^{-1})^3 + 5*(2 * r^{-1})^1 * (s^{-1})^4 + 1*(2 * r^{-1})^0 * (s^{-1})^5\]

Simplify the Expanded Expression

Simplify each term in the expanded expression. After simplification, our final expanded term is: \[ 32 r^{-5} + 80 r^{-4} s^{-1} + 80 r^{-3} s^{-2} + 40 r^{-2} s^{-3} + 10 r^{-1} s^{-4} + s^{-5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem

Understanding the binomial theorem is key to expanding expressions like $(a+b)^n$. This theorem provides a powerful method to expand binomials raised to any power, by expressing them as a sum of terms. Each term in the expansion corresponds to a combination of the binomial coefficients and the powers of each binomial part. The formula for the binomial theorem is:

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

Here, $\binom{n}{k}$ is a binomial coefficient, $a^{n-k}$ and $b^k$ are the terms raised to the respective powers. This provides a structured approach to breaking down the expression into more manageable parts, especially useful when the powers are high, like in our example with power 5.

Binomial Coefficients

Binomial coefficients play a central role in the binomial expansion process. They determine how each term in the expanded form is scaled. The binomial coefficient $\binom{n}{k}$ represents the number of ways to choose $k$ elements from $n$, calculated using the formula:

$\binom{n}{k} = \frac{n!}{(n-k)!k!}$

These coefficients are symmetrical, which can often simplify calculations. In our exercise, for $(2r^{-1} + s^{-1})^5$, the binomial coefficients are:

$\binom{5}{0} = 1$
$\binom{5}{1} = 5$
$\binom{5}{2} = 10$
$\binom{5}{3} = 10$
$\binom{5}{4} = 5$
$\binom{5}{5} = 1$

These coefficients provide the numeric multipliers that, when combined with the terms, construct the full expansion.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent or power. It's essential when applying the binomial theorem, as each term's component must be raised to a particular power.

When expanding, you break down the powers of each expression, such as $(2r^{-1})^k$ and $(s^{-1})^{n-k}$.

This involves repeating multiplication, e.g., raising $2r^{-1}$ to the fifth power results in $(2^5)(r^{-1})^5 = 32r^{-5}$. Likewise, powers of $s^{-1}$ are computed similarly. Understanding this part well helps you effectively navigate and simplify the overall expansion.

Mathematical Simplification

After expanding a binomial using coefficients and exponents, simplification is crucial. It transforms a complex expression into a simpler form that is more readable and useful. Effective simplification involves:

Combining like terms, which in our exercise employs negative exponents, e.g., $r^{-1}$ and $s^{-1}$.
Handling coefficient multiplication, where each term combines numerically, keeping track of the powers of each variable.

In our expanded equation $32r^{-5} + 80r^{-4}s^{-1} + 80r^{-3}s^{-2} + 40r^{-2}s^{-3} + 10r^{-1}s^{-4} + s^{-5}$, each term has been simplified by maintaining the powers of $r$ and $s$ while rendering the coefficients straightforward, demonstrating the practical simplification of exponential terms.

91影视

Expand the binomial. $$\left(2 r^{-1}+s^{-1}\right)^{5}$$

Short Answer

Step by step solution

Identify the Binomial Coefficients

Calculate the Binomial Coefficients

Apply the Binomial Theorem

Simplify the Expanded Expression

Key Concepts

Binomial Theorem

Binomial Coefficients

Exponentiation

Mathematical Simplification

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