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This module is included inLens: Siyavula: Mathematics (Gr. 10-12)
By: Siyavula

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## Introduction

In Grade 10 we studied exponential numbers and learnt that there were six laws that made working with exponential numbers easier. There is one law that we did not study in Grade 10. This will be described here.

## Laws of Exponents

In Grade 10, we worked only with indices that were integers. What happens when the index is not an integer, but is a rational number? This leads us to the final law of exponents,

a m n = a m n a m n = a m n
(1)

### Exponential Law 7: amn=amnamn=amn

We say that xx is an nnth root of bb if xn=bxn=b and we write x=bnx=bn. n th n th roots written with the radical symbol, , are referred to as surds. For example, (-1)4=1(-1)4=1, so -1-1 is a 4th root of 1. Using law 6, we notice that

( a m n ) n = a m n × n = a m ( a m n ) n = a m n × n = a m
(2)

therefore amnamn must be an nnth root of amam. We can therefore say

a m n = a m n a m n = a m n
(3)

For example,

2 2 3 = 2 2 3 2 2 3 = 2 2 3
(4)

A number may not always have a real nnth root. For example, if n=2n=2 and a=-1a=-1, then there is no real number such that x2=-1x2=-1 because x20x20 for all real numbers xx.

#### Complex Numbers

There are numbers which can solve problems like x2=-1x2=-1, but they are beyond the scope of this book. They are called complex numbers.

It is also possible for more than one nnth root of a number to exist. For example, (-2)2=4(-2)2=4 and 22=422=4, so both -2 and 2 are 2nd (square) roots of 4. Usually, if there is more than one root, we choose the positive real solution and move on.

#### Exercise 1: Rational Exponents

Simplify without using a calculator:

5 4 - 1 - 9 - 1 1 2 5 4 - 1 - 9 - 1 1 2
(5)
##### Solution
1. Step 1. Rewrite negative exponents as numbers with positive indices :
= 5 1 4 - 1 9 1 2 = 5 1 4 - 1 9 1 2
(6)
2. Step 2. Simplify inside brackets :
= 5 1 ÷ 9 - 4 36 1 2 = 5 1 × 36 5 1 2 = ( 6 2 ) 1 2 = 5 1 ÷ 9 - 4 36 1 2 = 5 1 × 36 5 1 2 = ( 6 2 ) 1 2
(7)
3. Step 3. Apply exponential law 6 :
= 6 = 6
(8)

#### Exercise 2: More rational Exponents

Simplify:

( 16 x 4 ) 3 4 ( 16 x 4 ) 3 4
(9)
##### Solution
1. Step 1. Convert the number coefficient to index-form with a prime base :
= ( 2 4 x 4 ) 3 4 = ( 2 4 x 4 ) 3 4
(10)
2. Step 2. Apply exponential laws :
= 2 4 × 3 4 . x 4 × 3 4 = 2 3 . x 3 = 8 x 3 = 2 4 × 3 4 . x 4 × 3 4 = 2 3 . x 3 = 8 x 3
(11)

#### Applying laws

Use all the laws to:

1. Simplify:
 (a) (x0)+5x0-(0,25)-0,5+823(x0)+5x0-(0,25)-0,5+823 (b) s12÷s13s12÷s13 (c) 12m798m-11912m798m-119 (d) (64m6)23(64m6)23
2. Re-write the following expression as a power of xx:
xxxxxxxxxx
(12)

## Exponentials in the Real-World

In Grade 10 Finance, you used exponentials to calculate different types of interest, for example on a savings account or on a loan and compound growth.

### Exercise 3: Exponentials in the Real world

A type of bacteria has a very high exponential growth rate at 80% every hour. If there are 10 bacteria, determine how many there will be in 5 hours, in 1 day and in 1 week?

#### Solution

1. Step 1. Population=Initialpopulation×(1+growthpercentage)timeperiodinhoursPopulation=Initialpopulation×(1+growthpercentage)timeperiodinhours :

Therefore, in this case:

Population=10(1,8)nPopulation=10(1,8)n, where nn = number of hours

2. Step 2. In 5 hours :

P o p u l a t i o n = 10 ( 1 , 8 ) 5 = 189 P o p u l a t i o n = 10 ( 1 , 8 ) 5 = 189

3. Step 3. In 1 day = 24 hours :

P o p u l a t i o n = 10 ( 1 , 8 ) 24 = 13 382 588 P o p u l a t i o n = 10 ( 1 , 8 ) 24 = 13 382 588

4. Step 4. in 1 week = 168 hours :

P o p u l a t i o n = 10 ( 1 , 8 ) 168 = 7 , 687 × 10 43 P o p u l a t i o n = 10 ( 1 , 8 ) 168 = 7 , 687 × 10 43

Note this answer is given in scientific notation as it is a very big number.

### Exercise 4: More Exponentials in the Real world

A species of extremely rare, deep water fish has an extremely long lifespan and rarely have children. If there are a total 821 of this type of fish and their growth rate is 2% each month, how many will there be in half of a year? What will the population be in 10 years and in 100 years?

#### Solution

1. Step 1. Population=Initialpopulation×(1+growthpercentage)timeperiodinmonthsPopulation=Initialpopulation×(1+growthpercentage)timeperiodinmonths :

Therefore, in this case:

Population=821(1,02)nPopulation=821(1,02)n, where nn = number of months

2. Step 2. In half a year = 6 months :

P o p u l a t i o n = 821 ( 1 , 02 ) 6 = 925 P o p u l a t i o n = 821 ( 1 , 02 ) 6 = 925

3. Step 3. In 10 years = 120 months :

P o p u l a t i o n = 821 ( 1 , 02 ) 120 = 8 838 P o p u l a t i o n = 821 ( 1 , 02 ) 120 = 8 838

4. Step 4. in 100 years = 1 200 months :

P o p u l a t i o n = 821 ( 1 , 02 ) 1 200 = 1 , 716 × 10 13 P o p u l a t i o n = 821 ( 1 , 02 ) 1 200 = 1 , 716 × 10 13

Note this answer is also given in scientific notation as it is a very big number.

## End of chapter Exercises

1. Simplify as far as possible:
1. 8-238-23
2. 16+8-2316+8-23
2. Simplify:
 (a) (x3)43(x3)43 (b) (s2)12(s2)12 (c) (m5)53(m5)53 (d) (-m2)43(-m2)43 (e) -(m2)43-(m2)43 (f) (3y43)4(3y43)4
3. Simplify as much as you can:
3a-2b15c-5a-4b3c-523a-2b15c-5a-4b3c-52
(13)
4. Simplify as much as you can:
9a6b4129a6b412
(14)
5. Simplify as much as you can:
a32b3416a32b3416
(15)
6. Simplify:
x3xx3x
(16)
7. Simplify:
x4b53x4b53
(17)
8. Re-write the following expression as a power of xx:
xxxxxx3xxxxxx3
(18)

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