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Chapter 9: Problem 61

Expanding an Expression In Exercises \(61-66,\) use the Binomial Theorem to expand and simplify the expression. $$(\sqrt{x}+5)^{3}$$

Short Answer

Expert verified
The expanded and simplified form of the expression \((\sqrt{x} + 5)^{3}\) using Binomial Theorem is \(x^{3/2} + 15x + 75\sqrt{x} + 125\).

Step by step solution

01

Recognizing the binomial

Recognize that \((\sqrt{x}+5)^3\) is a binomial in the form \((a+b)^n\), Here, \(a = \sqrt{x}\), \(b = 5\), and \(n = 3\).
02

Application of Binomial Theorem

Applying the Binomial Theorem, we expand the given expression by taking terms one by one as per the rule. Note that the binomial coefficient, \(n choose k\) can be calculated using the formula \(n!/(k!(n-k)!)\), where \(n!\) denotes the factorial of \(n\), multiplying all positive integers up to \(n\).
03

Expansion and simplification

We substitute the corresponding values for each term: Term 1: \(3 choose 0 * (\sqrt{x})^{3-0} * 5^0\) = \(3 choose 0 * x^{3/2} * 1\) Term 2: \(3 choose 1 * (\sqrt{x})^{3-1} * 5^1\) = \(3 choose 1 * x * 5\) Term 3: \(3 choose 2 * (\sqrt{x})^{3-2} * 5^2\) = \(3 choose 2 * \sqrt{x} * 25\) Term 4: \(3 choose 3 * (\sqrt{x})^{3-3} * 5^3\) = \(3 choose 3 * 1 * 125\) Now calculating the binomial coefficients and simplifying: Term 1: 1*\(x^{3/2}\) = \(x^{3/2}\) Term 2: 3*\(5x\) = \(15x\) Term 3: 3*\(25\sqrt{x}\) = \(75\sqrt{x}\) Term 4: 1*125 = 125 Adding all these terms, we get a simplified form of the expression.

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Key Concepts

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

Expression Expansion
Expression expansion involves breaking down an expression into its simpler components. In this problem, we start with \((\sqrt{x} + 5)^3\). This expression is a binomial, meaning it consists of two terms.
To expand it, we use the Binomial Theorem, which helps break down and express the powers of a binomial as a sum of terms.
Each term represents all possible combinations of the components raised to different powers. In simple terms, we're spreading out the expression to see its full mathematical structure.
Factoring Binomials
Factoring binomials is about expressing a sum or difference of two terms as a product. In this exercise, we don't directly factor, but understanding the binomial form \( (a + b)^n \) is essential.
It's crucial to figure out the values for \(a\), \(b\), and \(n\). Here, \(a = \sqrt{x}\), \(b = 5\), and \(n = 3\).
Instead of breaking it down into factors, we're multiplying it out into a longer sum. This step prepares the expression for expansion, helping us to systematically convert it using the Binomial Theorem.
Exponents and Radicals
Exponents and radicals play an important role in this exercise. The symbol \(\sqrt{x}\) is a radical, which is another way of saying \(x^{1/2}\). When you see something like \(\sqrt{x}\), think of it as raising \(x\) to the half power.
This helps in expanding expressions since it's easier to handle powers in calculations and apply rules for exponents, like the Power of a Power rule.
In this context, expressing \(\sqrt{x}\) as \(x^{1/2}\) allows us to neatly multiply and manage powers when applying the theorem, making the expansion process clearer and more organized.
Binomial Coefficients
Binomial coefficients are the numbers in front of terms in a binomial expansion. They tell you how many ways you can choose k terms from \(n\) items, represented as \( \binom{n}{k} \).
For instance, in \((\sqrt{x} + 5)^3\), you use coefficients like \( \binom{3}{0} \), \( \binom{3}{1} \), \( \binom{3}{2} \), and \( \binom{3}{3} \).
These coefficients increase the overall value of each term based on its binomial position. To calculate these, use the formula:
  • \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
Understanding this allows each expanded term to accurately reflect the contribution of \(a\) and \(b\) after expansion.

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Most popular questions from this chapter

Expanding an Expression In Exercises \(61-66,\) use the Binomial Theorem to expand and simplify the expression. $$(3 \sqrt{t}+\sqrt[4]{t})^{4}$$

Probability In Exercises \(85-88,\) consider \(n\) independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure.". The probability of a success on each trial is \(p,\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. To find the probability that the sales representative in Exercise 87 makes four sales when the probability of a sale with any one customer is \(\frac{1}{2},\) evaluate the term 8 $$_{4}\left(\frac{1}{2}\right)^{4}\left(\frac{1}{2}\right)^{4}$$ in the expansion of \(\left(\frac{1}{2}+\frac{1}{2}\right)^{8}\)

American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered \(1-36,\) of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on \(\quad\) three consecutive spins.

You and a friend agree to meet at your favorite fast-food restaurant between \(5 : 00\) P.M. and \(6 : 00\) P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour?

Linear Model, Quadratic Model, or Neither? In Exercises \(61-68\) , write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. $$a_{1}=3$$ $$a_{n}=a_{n-1}-n$$
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