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Chapter 10: Problem 103

Simplify each expression. Assume all variables represent positive numbers. $$ 3(\sqrt{5 b}-\sqrt{3})^{2} $$

Short Answer

Expert verified
The simplified expression is \(15b - 6\sqrt{15b} + 9\).

Step by step solution

01

Apply the Binomial Squared Formula

Use the formula for the square of a binomial: \[(a-b)^2 = a^2 - 2ab + b^2\]Here, the expressions are \(a = \sqrt{5b}\) and \(b = \sqrt{3}\). First, identify \((\sqrt{5b})^2\), \(2 \times \sqrt{5b} \times \sqrt{3}\), and \((\sqrt{3})^2\).
02

Simplify the Squared Terms

Calculate each term in the binomial expansion. The first term, \((\sqrt{5b})^2 = 5b\), and the last term, \((\sqrt{3})^2 = 3\).
03

Simplify the Middle Term

Calculate the middle term: \[2\times \sqrt{5b} \times \sqrt{3} = 2\times \sqrt{15b}\].So the expression becomes: \[5b - 2\sqrt{15b} + 3\].
04

Distribute the Constant Outside the Brackets

Now incorporate the 3 outside the parentheses: \[3(5b - 2\sqrt{15b} + 3)\].Distribute 3 across each term inside the brackets.
05

Final Simplification

After distributing the 3, the expression becomes: \[15b - 6\sqrt{15b} + 9\].This is the simplified version of the given expression.

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a powerful algebraic tool for expanding expressions that are raised to a power. Specifically, it helps in expanding a binomial expression鈥攁n expression with two terms like
  • \((a-b)^2\).
This theorem is helpful because multiplying out these expressions manually can be time-consuming. Using the binomial expansion formula, \[(a-b)^2 = a^2 - 2ab + b^2\],we can simplify the process. In our exercise, \(a = \sqrt{5b}\) and \(b = \sqrt{3}\). By substituting these into the formula, we calculate:
  • \((\sqrt{5b})^2 = 5b\)
  • Middle term: \(2 \times \sqrt{5b} \times \sqrt{3} = 2\sqrt{15b}\)
  • Last term: \((\sqrt{3})^2 = 3\).
These calculations form the simplified expression \(5b - 2\sqrt{15b} + 3\).
Radical Expressions
Radical expressions involve roots, like square roots, cube roots, etc. In our problem, these expressions are part of
  • the given expression \(\sqrt{5b}\)
  • and \(\sqrt{3}\).
To work with radical expressions effectively, you need to understand how to simplify them, especially when they appear within other algebraic terms. Here, simplifying \((\sqrt{5b})^2\) turns it into \(5b\), the corresponding whole number expression. Similarly, \(2\times \sqrt{5b}\times \sqrt{3}\) simplifies into \(2\sqrt{15b}\). These transformations are possible due to the properties of exponents and roots, which allow us to manipulate and simplify these calculations efficiently.
Distributive Property
The Distributive Property is fundamental in algebra and helps simplify expressions by distributing a factor across terms inside parentheses. In our exercise, after using the binomial theorem and radical simplifications, the expression becomes \(5b - 2\sqrt{15b} + 3\). The next step is applying the distributive property to the expression \(3(5b - 2\sqrt{15b} + 3)\). Distribute the 3 across each term:
  • First term: \(3 \times 5b = 15b\)
  • Second term: \(3 \times (-2\sqrt{15b}) = -6\sqrt{15b}\)
  • Third term: \(3 \times 3 = 9\).
This results in the final simplified expression: \(15b - 6\sqrt{15b} + 9\). This method effectively turns a complex term into a simplified, easy-to-work-with expression, showcasing the distributive property's usefulness.

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Most popular questions from this chapter

The illustration shows the graph of the quadratic function \(f(x)=-4 x^{2}+12 x\) with domain \([0,3] .\) Explain how the value of \(f(x)\) changes as the value of \(x\) increases from 0 to 3.

Suppose you are a police patrol officer and you have a 300 -foot-long roll of yellow "DO NOT CROSS" barricade tape to seal off an automobile accident, as shown in the illustration. What dimensions should you use to seal off the maximum rectangular area around the collision? What is the maximum area?

The height \(h\) (in feet) of an object that is dropped from a height of \(s\) feet is given by the formula \(h=s-16 t^{2}\) where \(t\) is the time the object has been falling. A 5 -foot-tall woman on a sidewalk looks directly overhead and sees a window washer drop a bottle from four stories up. How long does she have to get out of the way? Round to the nearest tenth. (A story is 12 feet.)

Explain the steps involved in expressing \(8 \pm \frac{\sqrt{15}}{2}\) as a single fraction with denominator 2.

WRITING Describe the following solution set of a rational inequality in words: \((-\infty, 4] \cup(6,7)\)
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