/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Expand the expression. $$ (2... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 49

Expand the expression. $$ (2+\sqrt{3})^{4} $$

Short Answer

Expert verified
The short answer is: \((2+\sqrt{3})^{4} = 97 + 56\sqrt{3}\)

Step by step solution

01

Identify the values of a, b, and n in the binomial theorem

In our case, \(a = 2\), \(b = \sqrt{3}\), and \(n = 4\), as given in the initial expression \((2+\sqrt{3})^4\).
02

Use the binomial theorem to get the terms of the expanded form

Using the binomial theorem, let's find the individual terms by applying the formula \({n \choose k} a^{n-k} b^k\): \(k = 0\): \({4 \choose 0} (2)^{(4-0)} (\sqrt{3})^{0} \) \\ \(k = 1\): \({4 \choose 1} (2)^{(4-1)} (\sqrt{3})^{1} \) \\ \(k = 2\): \({4 \choose 2} (2)^{(4-2)} (\sqrt{3})^{2} \) \\ \(k = 3\): \({4 \choose 3} (2)^{(4-3)} (\sqrt{3})^{3} \) \\ \(k = 4\): \({4 \choose 4} (2)^{(4-4)} (\sqrt{3})^{4} \) Now, let's calculate the individual terms.
03

Calculate individual terms

Calculating the terms, we get: \[\begin{array}{l} {4 \choose 0} (2)^{(4-0)} (\sqrt{3})^{0} = 1 \cdot 16 \cdot 1 = 16 \\ {4 \choose 1} (2)^{(4-1)} (\sqrt{3})^{1} = 4 \cdot 8 \cdot \sqrt{3} = 32\sqrt{3} \\ {4 \choose 2} (2)^{(4-2)} (\sqrt{3})^{2} = 6 \cdot 4 \cdot 3 = 72 \\ {4 \choose 3} (2)^{(4-3)} (\sqrt{3})^{3} = 4 \cdot 2 \cdot 3\sqrt{3} = 24\sqrt{3} \\ {4 \choose 4} (2)^{(4-4)} (\sqrt{3})^{4} = 1 \cdot 1 \cdot 9 = 9 \end{array}\]
04

Add the calculated terms to find the expanded form

Now, adding the calculated terms, we can write the expanded form of the given expression: \((2+\sqrt{3})^{4} = 16 + 32\sqrt{3} + 72 + 24\sqrt{3} + 9\)
05

Simplify the expanded form

We can simplify the expression by combining the constants and the terms containing the square root: \((2+\sqrt{3})^{4} = (16 + 72 + 9) + (32\sqrt{3} + 24\sqrt{3}) \) \((2+\sqrt{3})^{4} = 97 + 56\sqrt{3}\) So, the expanded form of \((2+\sqrt{3})^{4}\) is \(97 + 56\sqrt{3}\).

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