/*! 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 50 Expand the expression. $$ (3... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 50

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

Short Answer

Expert verified
The expanded expression of \((3+\sqrt{2})^4\) is: \[ (3+\sqrt{2})^4 = 197 + 108\sqrt{2} + 12(\sqrt{2})^3 \]

Step by step solution

01

Identify the binomial theorem formula

The binomial theorem states that for any real numbers 'a' and 'b' and non-negative integer 'n': \[ (a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k \] where \(\sum\) denotes the sum of the terms generated by the formula, and \({n \choose k}\) is the binomial coefficient. We can calculate the binomial coefficients using the formula: \[ {n \choose k} = \frac{n!}{k!(n-k)!} \] where 'n!' denotes the factorial of 'n' - the product of all positive integers up to 'n'.
02

Applying the binomial theorem formula to the given expression

In our case, \(a = 3\), \(b = \sqrt{2}\), and \(n = 4\). Using the binomial theorem formula, we get: \[ (3+\sqrt{2})^4 = \sum_{k=0}^4 {4 \choose k} 3^{4-k} (\sqrt{2})^k \]
03

Calculate the binomial coefficients

We need to calculate the binomial coefficients \({4 \choose 0}\), \({4 \choose 1}\), \({4 \choose 2}\), \({4 \choose 3}\), and \({4 \choose 4}\). Using the binomial coefficient formula: \[ {4 \choose 0} = \frac{4!}{0!4!} = 1 \] \[ {4 \choose 1} = \frac{4!}{1!3!} = 4 \] \[ {4 \choose 2} = \frac{4!}{2!2!} = 6 \] \[ {4 \choose 3} = \frac{4!}{3!1!} = 4 \] \[ {4 \choose 4} = \frac{4!}{4!0!} = 1 \]
04

Calculate each term of the expression

Now, we use both the binomial coefficients and the original formula to expand the expression: \[ (3+\sqrt{2})^4 = 1 \cdot 3^4 \cdot (\sqrt{2})^0 +4 \cdot 3^3 \cdot (\sqrt{2})^1 +6 \cdot 3^2 \cdot (\sqrt{2})^2 +4 \cdot 3^1 \cdot (\sqrt{2})^3 +1 \cdot 3^0 \cdot (\sqrt{2})^4 \]
05

Simplify each term

We will now simplify each term of the expression to obtain the result: \[ (3+\sqrt{2})^4 = 1 \cdot 81 \cdot 1 +4 \cdot 27 \cdot \sqrt{2} +6 \cdot 9 \cdot 2 +4 \cdot 3 \cdot (\sqrt{2})^3 +1 \cdot 1 \cdot 8 \]
06

Compute the sum

Finally, we compute the sum of all the terms: \[ (3+\sqrt{2})^4 = 81 + 108\sqrt{2} + 108 + 12(\sqrt{2})^3 + 8 \]
07

Write the final result

The expanded expression of \((3+\sqrt{2})^4\) is: \[ (3+\sqrt{2})^4 = 197 + 108\sqrt{2} + 12(\sqrt{2})^3 \]

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Most popular questions from this chapter

Suppose \(t(x)=\frac{5}{4 x^{3}+3}\). (a) Show that the point (-1,-5) is on the graph of \(t\) (b) Give an estimate for the slope of a line containing (-1,-5) and a point on the graph of \(t\) very close to (-1,-5)

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