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Chapter 13: Problem 96

\(\sqrt{8}\)

Short Answer

Expert verified
\( \sqrt{8} = 2 \sqrt{2} \)

Step by step solution

01

Identify the problem

The exercise requires finding the simplified form of the square root \( \sqrt{8} \).
02

Factorize the number under the radical

Factorize 8 into its prime factors: 8 = 4 \times 2. Note that 4 is 2 squared, so the expression can be written as \( 2^2 \times 2 \).
03

Apply the square root to the factors

Apply the square root to each factor: \( \sqrt{8} = \sqrt{2^2 \times 2} \). According to the properties of square roots, this can be expressed as \( \sqrt{2^2 \times 2} = \sqrt{2^2} \times \sqrt{2} \).
04

Simplify the square roots

Simplify the expression: \( \sqrt{2^2} \times \sqrt{2} = 2 \times \sqrt{2} \).
05

Write the final simplified form

Combine the terms to get the simplified form: \( \sqrt{8} = 2 \sqrt{2} \).

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Key Concepts

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

Prime Factorization
Before diving into simplification, it's crucial to understand prime factorization. Prime factorization means breaking down a number into its prime factors鈥攚hich are numbers that are greater than 1 and only divisible by 1 and themselves.
For the number 8, we can break it down into prime factors: 8 = 2 脳 2 脳 2 or in exponential form, 8 = 2^3.
Identifying the prime factors helps in simplifying square roots because it reveals if any factors can be 'paired'鈥攐r expressed as squares鈥攚hich can be easily simplified under the radical symbol.
Properties of Square Roots
Next, it's essential to understand some key properties of square roots to simplify expressions efficiently. One important property is that the square root of a product is the product of the square roots:
\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
Applying this property can make complex-looking problems much simpler.
In the case of our example from the exercise, \ \( \sqrt{8} \), we have broken down 8 into prime factors: \ \( 8 = 2^2 \times 2 \).
Using the property of square roots, we can express \ \( \sqrt{8} \) as: \ \( \sqrt{2^2 \times 2} = \sqrt{2^2} \times \sqrt{2} \).
Simplified Form
Finally, let's combine what we've learned to achieve the simplified form. By applying our knowledge of prime factorization and square root properties, we can simplify \ \( \sqrt{8}\).
We already broke down 8 into its prime factors and applied the square root to each factor. We found that: \sqrt{2^2 \times 2} = \sqrt{2^2} \times \sqrt{2} = 2 \times \sqrt{2} \.
Therefore, the simplified form of \ \( \sqrt{8} \) ends up being: \ \( 2 \sqrt{2} \). Simplification makes square roots easier to handle in more complex mathematical problems.
Ensure you always break factors down fully and look for pairs that can simplify under the radical symbol.

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Most popular questions from this chapter

Suppose you were offered a job in which you would work 8 hours per day for 5 workdays per week for 1 month at hard manual labor. Your pay the first day would be 1 penny. On the second day your pay would be two pennies; the third day 4 pennies. Your pay would double on each successive workday. There are 22 workdays in the month. There will be no sick days. If you miss a day of work, there is no pay or pay increase. How much do you get paid if you work all 22 days? How much do you get paid for the 22nd workday? What risks do you run if you take this job offer? Would you take the job?

Suppose \(x, y, z\) are consecutive terms in a geometric sequence. If \(x+y+z=103\) and \(x^{2}+y^{2}+z^{2}=6901,\) find the value of \(y\)

Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points. $$ \left\\{\begin{array}{r} x \geq 0 \\ y \geq 0 \\ x+y \leq 6 \\ 2 x+y \leq 10 \end{array}\right. $$

Bode's Law In \(1772,\) Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun: $$ a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2} $$ where \(n \geq 2\) is the number of the planet from the sun. (a) Determine the first eight terms of the sequence. (b) At the time of Bode's publication, the known planets were Mercury \((0.39 \mathrm{AU}),\) Venus \((0.72 \mathrm{AU}),\) Earth \((1 \mathrm{AU})\) Mars \((1.52 \mathrm{AU}),\) Jupiter \((5.20 \mathrm{AU}),\) and Saturn \((9.54 \mathrm{AU})\) How do the actual distances compare to the terms of the sequence? (c) The planet Uranus was discovered in \(1781,\) and the asteroid Ceres was discovered in \(1801 .\) The mean orbital distances from the sun to Uranus and Ceres " are \(19.2 \mathrm{AU}\) and \(2.77 \mathrm{AU},\) respectively. How well do these values fit within the sequence? (d) Determine the ninth and tenth terms of Bode's sequence. (e) The planets Neptune and Pluto" were discovered in 1846 and \(1930,\) respectively. Their mean orbital distances from the sun are \(30.07 \mathrm{AU}\) and \(39.44 \mathrm{AU},\) respectively. How do these actual distances compare to the terms of the sequence? (f) On July \(29,2005,\) NASA announced the discovery of a dwarf planet \((n=11),\) which has been named Eris. Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 8+4+2+\cdots $$
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