The Laws of exponents Study Guide…page 1 of 3
Base Number: The number that multiplies by itself as many times as the exponent tells it to. Exponent: The small number that tells the base number how many times to multiply by itself. NOTE: numbers and variables without exponents actually have an invisible 1 as their exponent.

| | | |Multiplying exponents |EX. 22 ( 23 | |with the same base number |long way | | |2(2 ( 2(2(2 = 25 = 32 | |1.) Keep the base number. | | | |short way | |2.) Add the exponents. |22 ( 23 = 22+3 = 25 = 32 | | | | |Multiplying exponents |EX. 4x3 ( -2x4 | |with variables |long way | | |4(x(x(x ( -2(x(x(x(x= -8x7 | |1.) Multiply the base numbers. | | | |short way | |2.) Add the exponents. |4x3 ( -2x4 = -8x3+4 = -8x7 | | | | |Raising a power to a power |EX. (32 xy4 )3 | | | | |1.) Multiply the exponent |long way | |on the outside of the ( ) |(32 xy4 ) (32 xy4 ) (32 xy4 ) | |by each exponent on the | | |inside. |(3(3(x(y(y(y(y) (3(3(x(y(y(y(y) (3(3(x(y(y(y(y) | | | | |2.) Simplify the base number. |36 x3 y12 = 729 x3 y12 | | |short way...

...Laws of Exponent
Lesson 1
Rules of 1
Any number raised to 1 is equal to the number itself
x¹=x
Examples: Common Error:
1.) 4¹= 4 1.) 4¹=4
2.) 5¹= 5 2.) 5¹=5
3.) 146¹=146 3.) 146¹=146
Practice Your Skill!
1.) 391¹=
2.) 45¹=
3.) 678¹=
4.) 99¹=
5.) 34¹=
Lesson 2
Product Rule
To multiply two powers having the same base, add their exponents.
xᵐ * xᵃ = xᵐᶧᵃ
Examples: Common Error:
1.)a² * a¹ = a³ 1.) a² * a¹= 2a⁶
2.) 5x²yz³ * 4xy³z² = 5*4x²ᶧ¹ + y¹ᶧ³ + z²ᶧ²= 20x³y⁴z⁵ 2.) 5x²yz³ * 4xy³z² = 9x²y³z⁶
Practice Your Skill!
Simplify the following expressions:
1.) 3y²*4y*3y³=
2.) 78x²y * -9y² =
3.) 45b⁸*11b⁷ =
Lesson 3
Power Rule
To raise a power, multiply the exponents
(xᵐ)ᵒ=xᵐᵒ
Examples: Common Error:
1.) (ab)³= a³b³ 1.) (ab)³= a¹ᶧ³b¹ᶧ³= a⁴b⁴
2.)...

...Laws of ExponentsExponents are also called Powers or Indices
The exponent of a number says how many times to use the number in a multiplication.
In this example: 82 = 8 × 8 = 64
In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"
.
So an Exponent just saves you writing out lots of multiplies!
Example: a7
a7 = a × a × a × a × a × a × a = aaaaaaa
Notice how I just wrote the letters together to mean multiply? We will do that a lot here.
Example: x6 = xxxxxx
The Key to the Laws
Writing all the letters down is the key to understanding the Laws
Example: x2x3 = (xx)(xxx) = xxxxx = x5
Which shows that x2x3 = x5, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.
All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:
The exponent says how many times to use the number in a multiplication.
A negative exponent means divide, because the opposite of multiplying is dividing
A fractional exponent like 1/n means to take the nth root:
If you understand those, then you understand exponents!
And all the laws below are based on those ideas....

...Laws of Exponents
Here are the Laws (explanations follow):
Law | Example |
x1 = x | 61 = 6 |
x0 = 1 | 70 = 1 |
x-1 = 1/x | 4-1 = 1/4 |
| |
xmxn = xm+n | x2x3 = x2+3 = x5 |
xm/xn = xm-n | x6/x2 = x6-2 = x4 |
(xm)n = xmn | (x2)3 = x2×3 = x6 |
(xy)n = xnyn | (xy)3 = x3y3 |
(x/y)n = xn/yn | (x/y)2 = x2 / y2 |
x-n = 1/xn | x-3 = 1/x3 |
And the law about Fractional Exponents: |
| |Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this example:
Example: Powers of 5 |
| .. etc.. | | |
52 | 1 × 5 × 5 | 25 | |
51 | 1 × 5 | 5 | |
50 | 1 | 1 | |
5-1 | 1 ÷ 5 | 0.2 | |
5-2 | 1 ÷ 5 ÷ 5 | 0.04 | |
| .. etc.. | | |
You will see that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or smaller) depending on whether the exponent gets larger (or smaller).
The law that xmxn = xm+n
With xmxn, how many times will you end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.
Example: x2x3 = (xx)(xxx) = xxxxx = x5
So, x2x3 = x(2+3) = x5
The law that xm/xn = xm-n
Like the previous example, how many times will you end up multiplying "x"? Answer: "m" times, then reduce...

...Laws of Exponents
Here are the Laws (explanations follow):
Law | Example |
x1 = x | 61 = 6 |
x0 = 1 | 70 = 1 |
x-1 = 1/x | 4-1 = 1/4 |
| |
xmxn = xm+n | x2x3 = x2+3 = x5 |
xm/xn = xm-n | x6/x2 = x6-2 = x4 |
(xm)n = xmn | (x2)3 = x2×3 = x6 |
(xy)n = xnyn | (xy)3 = x3y3 |
(x/y)n = xn/yn | (x/y)2 = x2 / y2 |
x-n = 1/xn | x-3 = 1/x3 |
And the law about Fractional Exponents: |
| |Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:
The law that xmxn = xm+n
With xmxn, how many times will you end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.
Example: x2x3 = (xx)(xxx) = xxxxx = x5
So, x2x3 = x(2+3) = x5
The law that xm/xn = xm-n
Like the previous example, how many times will you end up multiplying "x"? Answer: "m" times, thenreduce that by "n" times (because you are dividing), for a total of "m-n" times.
Example: x4/x2 = (xxxx) / (xx) = xx = x2
So, x4/x2 = x(4-2) = x2
(Remember that x/x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)
This law can also show you why x0=1 :
Example: x2/x2 = x2-2 = x0 =1
The law that (xm)n = xmn
First you multiply "m" times. Then...

...TASK 1
Explain the reference to legal principle and relevant case law, the legal aspect of placing the ‘Klick’ clock in the shop window with a price tag attached.
Ann antiques has a rare ‘Klick’ clock on its shop with price tags of €1,000 attached. In spite of its wording the sign in the window does not constitute a legal offer, it is merely an invitation to treat. Invitation to treat is an indication that the person who invite is willing to enter into a negotiation but it is not yet prepared to be bound. This case may be seen in Fisher v Bell (1961). It was held that having switch-blade knives in the window of a shop was not the same as offering them for sale.
TASK 2
Analyze the reference to legal principle and relevant because law, the legal effect of the event that transpired between Ann and Beth ignoring the conversation that took place between Carol and Beth and advice as to whether the valid contract exist between them.
The original invitation to treat at €1,000 was met by an offer from Beth which offers €500 on the ‘Klick’ clock. After Ann received an offer from Beth, Ann made a counter offer on the clock that she would sell €750 for it. It is up to Beth to decide whether to accept the offer or not. A counter offer arises when the offeree tries to change the terms of an original offer.
For example, the Hyde v Wrench (1940) case. In that case, on 6th June, Wrench offered to sell his estate to Hyde for £1,000 but...

...This judgment is subject to final editorial corrections approved by the
court and/or redaction pursuant to the publisher’s duty in compliance
with the law, for publication in LawNet and/or the Singapore Law
Reports.
BNJ (suing by her lawful father and litigation
representative, B)
v
SMRT Trains Ltd and another
[2013] SGHC 286
High Court — Suit No 432 of 2011
Vinodh Coomaraswamy JC (as he then was)
29–31 October 2012; 1–2, 5–9, 19–20 November 2012; 11 March 2013
Tort — Negligence — Breach of Duty
Tort — Occupier’s Liability — Who is an Occupier
Tort — Negligence — Res Ipsa Loquitur
Tort — Breach of Statutory Duty — Essential Factors
Contract — Contractual Terms — Implied Terms
31 December 2013
Judgment reserved
Vinodh Coomaraswamy J:
1
On 3 April 2011, a train coming into the Ang Mo Kio MRT station
(“AMK Station”) struck the plaintiff, causing her tragic and life-changing
injuries. She was then just fourteen years old. In these proceedings, she seeks
damages from two defendants for the injuries she suffered on that day. The
first defendant is SMRT Trains Ltd (“SMRT”). SMRT is a public transport
operator and holds the license to operate the mass rapid transit (“MRT”)
system along the North-South line. SMRT operates AMK Station and the train
which injured the plaintiff. The second defendant is the Land Transport
Authority of Singapore (“the LTA”). The LTA is a statutory board charged
BNJ v SMRT Trains Ltd...

...Contract Law – Formative Assessment
Alex would be suing Betty for a breach of contract. He would only succeed if he’s able to prove that a contract was in place. A contract can be defined as “a written or spoken agreement that is intended to be enforceable by law.” In order for it to be formed, agreement must take place and it can be broken down into two elements. Firstly, an offer. This can be described as an expression of willingness to contract on clear terms, with the intention that it will become a binding contract when it has been accepted. The second is acceptance, which can be defined as the unqualified expression of assent to the terms of an offer.
Betty placing an advertisement in the Ealing advertiser for her BMW is clearly an invitation to treat and not an offer. An invitation to treat can be described as “a mere declaration of willingness to enter into negotiations”: Partridge v Crittenden. The purpose of an invitation to treat is to invite offers. By advertising in the paper she is inviting people to make offers for her car.
On Monday when Alex makes an offer of £10 000, this can be considered as a counter offer. This is because he is changing the terms of agreement. It’s an offer because as explained the definition above, he intends on forming a binding contract by giving her £10 000 for the car; however it is initially rejected by Betty.
When Betty writes a letter to Alex saying that she’ll take £11 000, this is...

...SUBJECTS OF INTERNATIONAL LAW - STATES
I. Traditional Subjects of International Law
A. States
In addition to controlling territory, States have lawmaking and executive functions. States have full legal capacity, that is, they have the ability to be vested with rights and to incur obligations.
B. Insurgents
Insurgents are a destabilizing factor, which makes States reluctant to accept them, unless they show some of the attributes of sovereignty (e.g. control of a defined territory). Their existence is temporary; they either prevail and become a full-fledged state, or fail and disappear.
II. Modern Subjects of International Law
All new modern subjects of international law lack permanent and stable control over a territory. They have limited legal capacity (do not have a full spectrum of rights and obligations) and limited legal capacity to act (i.e. to enforce their rights).
A. International Organizations
B. National Liberation Movements
C. Individuals
III. Conditions for Statehood and the Role of Recognition
Unlike national systems, the international legal order lacks a set of detailed rules regarding the creation of states. However, such rules can be inferred from custom.
A. Conditions for Statehood
The Montevideo Convention of 1933 lays the traditional and most widely accepted criteria of statehood in international law. It states “The state as a person of...

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