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Chapter 4: Problem 2

Evaluate without a calculator: a. \(\sqrt{10,000}\) b. \(\sqrt{-25}\) c. \(625^{1 / 2}\) d. \(100^{1 / 2}\) e. \(\left(\frac{1}{9}\right)^{1 / 2}\) f. \(\left(\frac{625}{100}\right)^{1 / 2}\)

Short Answer

Expert verified
a. 100, b. not a real number, c. 25, d. 10, e. 1/3, f. 2.5

Step by step solution

01

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9.
02

- Evaluate \(\text{a. } \sqrt{10,000}\)

The square root of 10,000 is the number that, when multiplied by itself, gives 10,000. Knowing that 100 脳 100 = 10,000, we get \(\sqrt{10,000} = 100\).
03

- Evaluate \(\text{b. } \sqrt{-25}\)

The square root of a negative number is not a real number and cannot be evaluated within the real number system. Therefore, \(\sqrt{-25}\) is not a real number.
04

- Evaluate \(\text{c. } 625^{1/2}\)

Raising a number to the power of 1/2 is equivalent to finding its square root. Since \(\sqrt{625} = 25\), we have \(\text{c. } 625^{1/2} = 25\).
05

- Evaluate \(\text{d. } 100^{1/2}\)

Raising 100 to the power of 1/2 is the same as finding its square root. Since \(\sqrt{100} = 10\), we have \(\text{d. } 100^{1/2} = 10\).
06

- Evaluate \(\text{e. } \left(\frac{1}{9}\right)^{1/2}\)

Raising \(\frac{1}{9}\) to the power of 1/2 means finding the square root of \(\frac{1}{9}\). The square root of \(\frac{1}{9}\) is \(\frac{1}{3}\), so \(\text{e. } \left(\frac{1}{9}\right)^{1/2} = \frac{1}{3}\).
07

- Evaluate \(\text{f. } \left(\frac{625}{100}\right)^{1/2}\)

First, simplify \(\frac{625}{100} = 6.25\). Then, the square root of 6.25 is 2.5, so \(\text{f. } \left(\frac{625}{100}\right)^{1/2} = 2.5\).

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Key Concepts

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

Evaluating Square Roots
When we evaluate square roots, we're looking for a number that, when squared, gives back our original number. For example, \(\sqrt{9} = 3\), since multiplying 3 by itself (i.e., \(3 \times 3 = 9\)) returns the original number.

Here are a few steps to understand this concept better:
  • The square root of 10,000 is 100, because \(100 \times 100 = 10,000\).
  • If you try to find the square root of a negative number, such as \(\sqrt{-25}\), you get a result that isn't a real number because no real number squared will result in a negative.
  • Raising a number to the power of \(\frac{1}{2}\) is another way to find its square root.
    • For instance, \(625^{1/2} = 25\) because \(\sqrt{625} = 25\).
    • Similarly, \(100^{1/2} = 10\) because the square root of 100 is 10.
    • For a fraction like \left(\frac{1}{9}\right)^{1/2}\, the square root of \(\frac{1}{9}\) equals \(\frac{1}{3}\).
    • Lastly, a fraction like \left(\frac{625}{100}\right)^{1/2}\ simplifies to \sqrt{6.25} = 2.5\.
Real Numbers
Real numbers include all the numbers that can be found on the number line. This includes both positive and negative numbers, zero, fractions, and decimals.

It's important to know that the square root of some numbers falls within the real numbers, while others do not:
  • Positive numbers always have a real number as their square root. For example, \(\sqrt{10,000} = 100\).
  • Negative numbers do not have real square roots. For instance, \(\sqrt{-25}\) does not exist in the real number system.
  • Fractions can also fall within the real number system. For example, \(\sqrt{\frac{1}{9}} = \frac{1}{3}\).

Understanding these distinctions helps you effectively evaluate square roots and identify which results lie within the realm of real numbers.
Exponents
Exponents are a way to express repeated multiplication of the same number. The general form is \(a^n\) where 'a' is the base, and 'n' is the exponent.

In square root problems, exponents with a power of \(\frac{1}{2}\) specifically refer to square roots:
  • \

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Most popular questions from this chapter

Write each of the following in scientific notation: a. 0.00029 d. 0.00000000001 $$ \text { g. }-0.0049 $$ b. 654.456 e. 0.00000245 c. 720,000 $$ \text { f. }-1,980,000 $$

Rewrite in an equivalent form using logarithms: a. \(10^{4}=10.000\) b. \(10^{-2}=0.01\) c. \(10^{0}=1\) d. \(10^{-5}=0.00001\)

In the December 1999 issue of the journal Science, two Harvard scientists describe a pair of "nanotweezers" they created that are capable of manipulating objects as small as one- 50,000 th of an inch in width. The scientists used the tweezers to grab and pull clusters of polystyrene molecules, which are of the same size as structures inside cells. A future use of these nanotweezers may be to grab and move components of biological cells. a. Express one- 50,000 th of an inch in scientific notation. b. Express the size of objects the tweezers are able to manipulate in meters. c. The prefix "nano" refers to nine subdivisions by \(10,\) or a multiple of \(10^{-9}\). So a nanometer would be \(10^{-9}\) meters. Is the name for the tweezers given by the inventors appropriate? d. If not, how many orders of magnitude larger or smaller would the tweezers' ability to manipulate small objects have to be in order to grasp things of nanometer size?

a. In 2006 Japan had a population of approximately 127.5 million people and a total land area of about 152.5 thousand square miles. What was the population density (the number of people per square mile)? b. In 2006 the United States had a population of approximately 300 million people and a total land area of about 3620 thousand square miles. What was the population density of the United States? c. Compare the population densities of Japan and the United States.

Estimate the radius, \(r,\) of a circular region with an area, \(A,\) of \(35 \mathrm{ft}^{2}\) (where \(A=\pi r^{2}\) ).
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