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Laws of Exponents

As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

The following are generally referred to as the "laws" or "rules" of exponents.

x^a x^b = x^(a+b)

x^{a} x^{b} = x^{a + b}

(1)

\frac{x^{a}}{x^{b}} = x^{a - b} = \frac{1}{x^{b - a}}

(2)

(x^a)^b=x^(ab)

(3)

As with any formula, the most important thing is to be able to use them--that is, to understand what they mean. But it is also important to know where these formulae come from. And finally, in this case, the three should be memorized.

So...what do they mean? They are, of course, algebraic generalizations--statements that are true for any x

x

, a

a

, and b

b

values. For instance, the first rule tells us that

7^{12} \cdot 7^{4} = 7^{16}

which you can confirm on your calculator. 7¹² means twelve 7s multiplied; 7⁴ means four 7s multiplied. Multiply them together, and you get sixteen 7s multiplied.

You can also think of the second rule as the inevitable consequence of the first rule. If 19³ x 19⁵ = 19⁸, then

\frac{19^{8}}{19^{5}}

(which asks the question "19⁵ times what equals 19⁸?") must be 19³.

Similarly, the third rule promises us that (7^12)^4 = 7^48

{(7^{12})}^{4} = 7^{48}

These rules can be used to combine and simplify expressions.

ASCIIMath notation

In order to mark up the previous expression in ASCIIMath, you would have to write it as: (7^12)^4 = 7^48. The ^ symbol represents exponents in ASCIIMath.

Theoretical Perspectives on Education

Functionalism

Functionalists view education as one of the more important social institutions in a society. They contend that education contributes two kinds of functions: manifest (or primary) functions, which are the intended and visible functions of education; and latent (or secondary) functions, which are the hidden and unintended functions.

Manifest Functions

There are several major manifest functions associated with education. The first is socialization. Beginning in preschool and kindergarten, students are taught to practice various societal roles. The French sociologist Émile Durkheim (1858-1917), who established the academic discipline of sociology, characterized schools as "socialization agencies that teach children how to get along with others and prepare them for adult economic roles. All education consists in a continuous effort to impose on the child ways of seeing, feeling, and acting to which it would not spontaneously come." (Durkheim 1898)

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Figure 1:

The teacher's authority in the classroom is a way in which education fulfills the manifest functions of social control. (Photo courtesy of Tulane Public Relations/flickr)

Latent Functions

Education also fulfills latent functions (shown in the table) . As you well know, much goes on in a school that has little to do with formal education. For example, you might notice an attractive fellow student when he gives a particularly interesting answer in class—catching up with him and making a date speaks to the latent function of courtship fulfilled by exposure to a peer group in the educational setting.

Manifest and Latent Functions of Education

Table 1: According to functionalist theory, education contributes both manifest and latent functions.
Manifest Functions: Openly stated functions with intended goals	Latent Functions: Hidden, unstated functions with sometimes unintended consequences
Socialization	Courtship
Transmission of culture	Social networks
Social control	Working in groups
Social placement	Creation of generation gap
Cultural innovation	Political and social integration

Example 1 Grade Inflation: When Is an A Really a C?

Consider a large-city newspaper publisher. Ten years ago, when culling résumés for an entry-level copywriter, they were well assured that if they selected a grad with a GPA of 3.7 or higher, they'd have someone with the writing skills to contribute to the workplace on day one. But over the last few years, they've noticed that A-level students don't have the competency evident in the past. More and more, they find themselves in the position of educating new hires in abilities that, in the past, had been mastered during their education.

This story illustrates a growing concern referred to as grade inflation—a term used to describe the observation that the correspondence between letter grades and the achievements they reflect has been changing (in a downward direction) over time. Put simply, what used to be considered C-level, or average, now often earns a student a B, or even an A.

Section Quiz

Exercise 1

Which of the following is not a manifest function of education? Hint: The answer can be found in the "Manifest Functions" section.

Cultural innovation
Courtship
Social placement
Socialization

Solution

[ Hide Solution ]

Exercise 2

Because she plans on achieving success in marketing, Tammie is taking courses on managing social media. This is an example of ________.

cultural innovation
social control
social placement
socialization

Solution

[ Hide Solution ]

Further Research

The National Center for Fair & Open Testing (FairTest) is committed to ending the bias and other flaws seen in standardized testing. Their mission is to ensure that students, teachers, and schools are evaluated fairly. You can learn more about their mission, as well as the latest in news on test bias and fairness, at their website: http://fairtest.org/

Theoretical Perspectives on Education

Functionalism

Manifest Functions

Latent Functions

Conflict Theory

Conflict theorists do not believe that public schools reduce social inequality. Rather, they believe that the educational system reinforces and perpetuates social inequalities arising from differences in class, gender, race, and ethnicity. Where functionalists see education as serving a beneficial role, conflict theorists view it more negatively. To them, educational systems preserve the status quo and push people of lower status into obedience.

Feminist Theory

Feminist theory aims to understand the mechanisms and roots of gender inequality in education, as well as their societal repercussions. Like many other institutions of society, educational systems are characterized by unequal treatment and opportunity for women. Almost two-thirds of the world's 862 million illiterate people are women, and the illiteracy rate among women is expected to increase in many regions, especially in several African and Asian countries (UNESCO 2005; 'World Bank 2007).

Symbolic Interactionism

Symbolic interactionism sees education as one way labeling theory is seen in action. A symbolic interactionist might say that this labeling has a direct correlation to those who are in power and those who are being labeled. For example, low standardized test scores or poor performance in a particular class often lead to a student being labeled as a low achiever. Such labels are difficult to "shake off," which can create a self-fulfilling prophecy (Merton 1968).

Summary

The major sociological theories offer insight into how we understand education. Functionalists view education as an important social institution that contributes both manifest and latent functions. Functionalists see education as serving the needs of society by preparing students for later roles, or functions, in society. Conflict theorists see schools as a means for perpetuating class, racial-ethnic, and gender inequalities. In the same vein, feminist theory focuses specifically on the mechanisms and roots of gender inequality in education. The theory of symbolic interactionism focuses on education as a means for labeling individuals.

Signals

A signal's complexity is not related to how wiggly it is. Rather, a signal expert looks for ways of decomposing a given signal into a sum of simpler signals, which we term the signal decomposition. Though we will never compute a signal's complexity, it essentially equals the number of terms in its decomposition. In writing a signal as a sum of component signals, we can change the component signal's gain by multiplying it by a constant and by delaying it. More complicated decompositions could contain derivatives or integrals of simple signals. In short, signal decomposition amounts to thinking of the signal as the output of a linear system having simple signals as its inputs. We would build such a system, but envisioning the signal's components helps understand the signal's structure. Furthermore, you can readily compute a linear system's output to an input decomposed as a superposition of simple signals. x+sqrt(1-x^2)

ASCIIMath error: syntax error LaTeX error: sqrt requires initial backslash: \sqrt $x + \sqrt{1 - x^{2}}$

Example 1 Add a title (optional)

As an example of signal complexity, we can express the pulse p_Delta(t) Δ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow> <mrow><mrow><mi></mi><msub> <mi>p</mi> <mi>?</mi> </msub><mi></mi></mrow><mo></mo><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced></mrow> </mrow><annotation-xml encoding="MathML-Content"><apply> <ci type="fn"> <msub> <mi>p</mi> <mi>?</mi> </msub> </ci> <ci>t</ci> </apply></annotation-xml></semantics></math>$ as a sum of delayed unit steps.

Add a title (optional)

p_Delta(t) = u(t) - u(t-Delta) Δ Δ

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow> <mrow><mrow><mrow><mi></mi><msub> <mi>p</mi> <mi>?</mi> </msub><mi></mi></mrow><mo></mo><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced></mrow></mrow><mrow><mo>=</mo></mrow><mrow><mrow><mrow><mrow><mi>u</mi></mrow><mo></mo><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced></mrow><mo>-</mo><mrow><mrow><mi>u</mi></mrow><mo></mo><mfenced open="(" close=")" separators=","><mrow><mrow><mi>t</mi></mrow><mo>-</mo><mrow><mi>?</mi></mrow></mrow></mfenced></mrow></mrow></mrow> </mrow><annotation-xml encoding="MathML-Content"><apply> <eq/> <apply> <ci type="fn"> <msub> <mi>p</mi> <mi>?</mi> </msub> </ci> <ci>t</ci> </apply> <apply> <minus/> <apply> <ci type="fn">u</ci> <ci>t</ci> </apply> <apply> <ci type="fn">u</ci> <apply> <minus/> <ci>t</ci> <ci>?</ci> </apply> </apply> </apply> </apply></annotation-xml></semantics></math>$

(1)

Thus, the pulse is a more complex signal than the step. Be that as it may, the pulse is very useful to us.

Exercise 1 Add a title (optional)

Express a square wave having period T $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow> <mrow><mi>T</mi></mrow> </mrow><annotation-xml encoding="MathML-Content"><ci>T</ci></annotation-xml></semantics></math>$ and amplitude A $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow> <mrow><mi>A</mi></mrow> </mrow><annotation-xml encoding="MathML-Content"><ci>A</ci></annotation-xml></semantics></math>$ as a superposition of delayed and amplitude-scaled pulses.

Solution

sq(t) sum_(n=-infty)^infty (-1)^n A p_{p/2}{t-n{T/2}} Σ ∞ ∞ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow> <mrow><mrow><mrow><mi>sq</mi></mrow><mo></mo><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced></mrow><mo>=</mo><mrow><munderover><mo>?</mo><mrow><mi> <mrow><mi>n</mi></mrow> </mi><mi>=</mi><mrow><mo>-</mo><mi>?</mi></mrow></mrow><mrow><mi>?</mi></mrow></munderover><mrow><msup><mfenced separators=" "><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi></mrow></msup><mo></mo><mrow><mi>A</mi></mrow><mo></mo><mrow><mi> </mi><msub> <mi>p</mi> <mi>T/2</mi> </msub><mi> </mi></mrow><mo></mo><mrow><mo>(</mo><mrow><mi>t</mi></mrow><mo>-</mo><mrow><mrow><mi>n</mi></mrow><mo></mo><mfrac><mrow><mi>T</mi></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow> </mrow><annotation-xml encoding="MathML-Content"><apply> <eq/> <apply> <ci type="fn">sq</ci> <ci>t</ci> </apply> <apply> <sum/> <bvar> <ci>n</ci> </bvar> <lowlimit> <apply> <minus/> <infinity/> </apply> </lowlimit> <uplimit> <infinity/> </uplimit> <apply> <times/> <apply> <power/> <apply> <minus/> <cn>1</cn> </apply> <ci>n</ci> </apply> <ci>A</ci> <ci> <msub> <mi>p</mi> <mi>T/2</mi> </msub> </ci> <apply> <minus/> <ci>t</ci> <apply> <times/> <ci>n</ci> <apply> <divide/> <ci>T</ci> <cn>2</cn> </apply> </apply> </apply> </apply> </apply> </apply></annotation-xml></semantics></math>$

Theorem 1: Pythagorean Theorem

On a right triangle, the sum of the squares of the sides equals the square of the hypotenus.

Proof Add a title (optional)

Take a right triangle whose sides are of length 3, 4, and 5. In this case the sum of the square of the two shorter sides is 9+16=25. The square of the hypotenus is 25. So the theorem holds.

note:

The Pythagorean Theorem does not apply to acute and obtuse triangles. It can only be applied to right triangles. However, a corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute.

Example 2 Add a title (optional)

Add a title (optional)

5² + 7² < 9²

(2)

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Glossary

sidereal period:: A period determined by or from the stars.

OERPub

Advanced

Edit: Conversion of Cooking_with_spam.doc

Laws of Exponents

ASCIIMath notation

Theoretical Perspectives on Education