The following are rational numbers:
• Fractions with both denominator and numerator as integers.
• Integers.
• Decimal numbers that end.
3.8
End of Chapter Exercises
1. Ifais an integer,bis an integer andcis not an integer, which of the following are rational numbers:
(a) 56 (b) a3 (c) b2 (d) 1c
2. Write each decimal as a simple fraction: (a) 0,5
(b) 0,12
(c) 0,6
(d) 1,59
(e) 12,277
3. Show that the decimal3,2 ˙1˙8is a rational number.
4. Showing all working, express0,7 ˙8as a fraction ab wherea, b∈Z.
Chapter 4
Exponentials - Grade 10
4.1
Introduction
In this chapter, you will learn about the short-cuts to writing2×2×2×2. This is known as writing a number inexponential notation.
4.2
Definition
Exponential notation is a short way of writing the same number multiplied by itself many times. For example, instead of5×5×5, we write53to show that the number 5 is multiplied by itself
3 times and we say “5 to the power of 3”. Likewise52is5
×5and35is3
×3×3×3×3. We will now have a closer look at writing numbers using exponential notation.
Definition: Exponential Notation
Exponential notation means a number written like
an
whennis an integer andacan be any real number. ais called thebase andnis called the exponent.
Thenth power ofais defined as:
an= 1×a×a×. . .×a (n times) (4.1)
withamultiplied by itselfntimes.
We can also define what it means if we have a negative index,−n. Then,
a−n= 1÷a÷a÷. . .÷a (ntimes) (4.2)
Important: Exponentials
If n is an even integer, then an will always be positive for any non-zero real number a. For
example, although−2 is negative,(−2)2 = 1
× −2× −2 = 4 is positive and so is(−2)−2 =
1÷ −2÷ −2 = 14.
4.3
Laws of Exponents
There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been seen in earlier grades, but we will list all the laws here for easy reference, but we will explain each law in detail, so that you can understand them, and not only remember them. a0 = 1 (4.3) am ×an = am+n (4.4) a−n = 1 an (4.5) am÷an = am−n (4.6) (ab)n = anbn (4.7) (am)n = amn (4.8)
4.3.1
Exponential Law 1:
a
0= 1
Our definition of exponential notation shows that
a0 = 1,(a6= 0) (4.9)
For example,x0= 1and(1 000 000)0= 1.
Exercise: Application using Exponential Law 1: a0= 1,(a
6 = 0) 1. 160= 1 2. 16a0= 16 3. (16 +a)0= 1 4. (−16)0= 1 5. −160= −1
4.3.2
Exponential Law 2:
a
m×a
n=a
m+nOur definition of exponential notation shows that
am ×an = 1 ×a×. . .×a (mtimes) (4.10) ×1×a×. . .×a (ntimes) = 1×a×. . .×a (m+ntimes) = am+n For example, 27×23 = (2×2×2×2×2×2×2)×(2×2×2) = 210 = 27+3
Interesting Fact
teresting
Fact
This simple law is the reason why exponentials were originally invented. In thedays before calculators, all multiplication had to be done by hand with a pencil and a pad of paper. Multiplication takes a very long time to do and is very tedious. Adding numbers however, is very easy and quick to do. If you look at what this law is saying you will realise that it means that adding the exponents of two exponential numbers (of the same base) is the same as multiplying the two numbers together. This meant that for certain numbers, there was no need to actually multiply the numbers together in order to find out what their multiple was. This saved mathematicians a lot of time, which they could use to do something more productive.Exercise: Application using Exponential Law 2: am
×an =am+n
1. x2.x5=x7
2. 2x3y
×5x2y7= 10x5y8
3. 23.24= 27[Take note that the base (2) stays the same.]
4. 3×32a
×32= 32a+3
4.3.3
Exponential Law 3:
a
−n=
1an
, a6= 0
Our definition of exponential notation for a negative exponent shows that
a−n = 1÷a÷. . .÷a (ntimes) (4.11)
= 1
1×a× · · · ×a (ntimes)
= 1
an
This means that a minus sign in the exponent is just another way of writing that the whole exponential number is to be divided instead of multiplied.
For example,
2−7 = 1
2×2×2×2×2×2×2
= 1
27
Exercise: Application using Exponential Law 3: a−n= 1
an, a6= 0 1. 2−2= 1 22 = 1 4 2. 23−22 = 1 22.32 = 1 36 3. (23)−3= (3 2)3= 27 8 31
4. m n−4 =mn4 5. a−3 .x4 a5.x−2 = x4 .x2 a3.a5 = x6 a8
4.3.4
Exponential Law 4:
a
m÷a
n=a
m−nWe already realised with law 3 that a minus sign is another way of saying that the exponential number is to be divided instead of multiplied. Law 4 is just a more general way of saying the same thing. We can get this law by just multiplying law 3 byamon both sides and using law 2.
am an = a ma−n (4.12) = am−n For example, 27 ÷23 = 2×2×2×2×2×2×2 2×2×2 = 2×2×2×2 = 24 = 27−3
Exercise: Exponential Law 4: am÷an =am−n
1. aa62 =a
6−2=a4
2. 3326 = 32−6= 3−4=
1
34[Always give final answer with positive index] 3. 324aa82 = 8a−6= 8 a6 4. a3x a4 =a3x−4
4.3.5
Exponential Law 5:
(ab)
n=a
nb
nThe order in which two real numbers are multiplied together does not matter. Therefore,
(ab)n = a×b×a×b×. . .×a×b (ntimes) (4.13) = a×a×. . .×a (ntimes) ×b×b×. . .×b (ntimes) = anbn For example, (2·3)4 = (2·3)×(2·3)×(2·3)×(2·3) = (2×2×2×2)×(3×3×3×3) = (24)×(34) = 2434
Exercise: Exponential Law 5: (ab)n=anbn 1. (2x2y)3= 23x2×3y5= 8x6y5 2. (7a b3) 2= 49a2 b6 3. (5an−4)3= 125a3n−12
4.3.6
Exponential Law 6:
(a
m)
n=a
mnWe can find the exponential of an exponential just as well as we can for a number. After all, an exponential number is a real number.
(am)n = am×am×. . .×am (ntimes) (4.14) = a×a×. . .×a (m×ntimes) = amn For example, (22)3 = (22)×(22)×(22) = (2×2)×(2×2)×(2×2) = (26) = 2(2×3)
Exercise: Exponential Law 6: (am)n=amn
1. (x3)4=x12
2. [(a4)3]2=a24
3. (3n+3)2= 32n+6
Worked Example 1: Simlifying indices Question: Simplify: 52x−1
.9x−2
152x−3 Answer
Step 1 : Factorise all bases into prime factors:
= 5 2x−1.(32)x−2 (5.3)2x−3 = 5 2x−1.32x−4 52x−3.32x−3
Step 2 : Add and subtract the indices of the sam bases as per laws 2 and 4:
= 52x−1−2x−3.32x−4−2x+3 = 52.3−1
Step 3 : Write simplified answer with positive indices:
= 25 3
Activity :: Investigation : Exponential Numbers
Match the answers to the questions, by filling in the correct answer into the Answer column. Possible answers are: 32, 1, -1, -3, 8.
Question Answer 23 73−3 (2 3)− 1 87−6 (−3)−1 (−1)23
4.4
End of Chapter Exercises
1. Simplify as far as possible:
(a)3020 (b)10 (c)(xyz)0 (d)[(3x4y7z12)5(
−5x9y3z4)2]0
(e)(2x)3 (f)(
−2x)3 (g)(2x)4 (h)(
−2x)4
2. Simplify without using a calculator. Leave your answers with positive exponents. (a) 3x−3 (3x)2 (b) 5x0+ 8−2−(1 2)−2.1x (c) 5b−3 5b+1
3. Simplify, showing all steps: (a) 2a−2 .3a+3 6a (b) ama2+mn++np+.apm (c) 3n.9n−3 27n−1 (d) (2yx−2ba)3
(e) 23x−1
.8x+1
42x−2 (f) 62x.112x
222x−1.32x
4. Simplify, without using a calculator: (a) (−3)(−−3)3.−(−43)2.
(b) (3−1+ 2−1)−1
(c) 9n−811.227−3n−2n
(d) 23n4+23n.−8n2−3
Chapter 5
Estimating Surds - Grade 10
5.1
Introduction
You should known by now what thenth root of a number means. If the nth root of a number cannot be simplified to a rational number, we call it asurd. For example,√2and√3
6are surds, but√4is not a surd because it can be simplified to the rational number 2.
In this chapter we will only look at surds that look like √na, where ais any positive number, for
example√7or √3
5. It is very common fornto be2, so we usually do not write √2a. Instead we write the surd as just√a, which is much easier to read.
It is sometimes useful to know the approximate value of a surd without having to use a calculator. For example, we want to be able to guess where a surd like√3 is on the number line. So how do we know where surds lie on the number line? From a calculator we know that√3is equal to
1,73205.... It is easy to see that√3is above 1 and below 2. But to see this for other surds like
√
18without using a calculator, you must first understand the following fact:
Interesting Fact
teresting
Fact
Ifaand bare positive whole numbers, and a < b, thenn
√a < √n
b. (Challenge: Can you explain why?)
If you don’t believe this fact, check it for a few numbers to convince yourself it is true.
How do we use this fact to help us guess what√18is? Well, you can easily see that18<25? Using our rule, we also know that√18<√25. But we know that52 = 25so that √25 = 5.
Now it is easy to simplify to get√18<5. Now we have a better idea of what√18is.
Now we know that√18is less than 5, but this is only half the story. We can use the same trick again, but this time with 18 on the right-hand side. You will agree that16 <18. Using our rule again, we also know that√16<√18. But we know that 16 is a perfect square, so we can simplify√16to4, and so we get4<√18!
Can you see now that we now have shown that√18 is between 4 and 5? If we check on our calculator, we can see that√18 = 4,1231..., and we see that our idea was right! You will notice that our idea used perfect squares that were close to the number 18. We found the closest perfect square underneath 18, which was 42 = 16, and the closest perfect square above 18,
which was52= 25. Here is a quick summary of what a perfect square or cube is:
Interesting Fact
teresting
Fact
A perfect square is the number obtained when an integer is squared. For example,9 is a perfect square since 32= 9. Similarly, a perfect cube is a number which isthe cube of an integer. For example, 27 is a perfect cube, because 33= 27.
To make it easier to use our idea, we will create a list of some of the perfect squares and perfect cubes. The list is shown in Table 5.1.
Table 5.1: Some perfect squares and perfect cubes Integer Perfect Square Perfect Cube
0 0 0 1 1 1 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 10 100 1000
Similarly, when given the surd √3
52you should be able to tell that it lies somewhere between 3 and 4, because √3
27 = 3and √3
64 = 4 and 52 is between 27 and 64. In fact √3
52 = 3,73. . .
which is indeed between 3 and 4.