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## Laws of exponents

Exponential notation is a short way of writing the same number multiplied by itself many times. We will now have a closer look at writing numbers using exponential notation. Exponents can also be called indices.

For any real number $a$ and natural number $n$, we can write a multiplied by itself $n$ times as $a n$.

## Identity 1

1. $a n =a×a×a×⋯×a(ntimes)(a∈ℝ,n∈ℕ)$

2. $a 0 =1$ ($a≠0$ because $0 0$ is undefined)

3. $a -n =1 a n$ ($a≠0$ because $1 0$ is undefined)

Examples:

1. $3×3=3 2$

2. $5×5×5×5=5 4$

3. $p×p×p=p 3$

4. $(3 x ) 0 =1$

5. $2 -4 =1 2 4 =1 16$

6. $1 5 -x =5 x$

Notice that we always write the final answer with positive exponents.

Chapter introduction

## Laws of exponents

There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been done in earlier grades, but we list all the laws here for easy reference:

### Identity 2

• $a m ×a n =a m+n$

• $a m a n =a m-n$

• $(ab) n =a n b n$

• $a b n =a n b n$

• $(a m ) n =a mn$

where $a>0$, $b>0$ and $m,n∈ℤ$.

### Example 1: Applying the exponential laws

#### Question

Simplify:

1. $2 3x ×2 4x$

2. $12p 2 t 5 3pt 3$

3. $(3x) 2$

4. $(3 4 5 2 ) 3$

1. $2 3x ×2 4x =2 3x+4x =2 7x$

2. $12p 2 t 5 3pt 3 =4p (2-1) t (5-3) =4pt 2$

3. $(3x) 2 =3 2 x 2 =9x 2$

4. $(3 4 ×5 2 ) 3 =3 (4×3) ×5 (2×3) =3 12 ×5 6$

### Example 2: Exponential expressions

#### Question

Simplify: $2 2n ×4 n ×2 16 n$

##### Change the bases to prime numbers
$2 2n ×4 n ×2 16 n =2 2n ×(2 2 ) n ×2 1 (2 4 ) n$(1)
##### Simplify the exponents
$=2 2n ×2 2n ×2 1 2 4n =2 2n+2n+1 2 4n =2 4n+1 2 4n =2 4n+1-(4n) =2$(2)

### Example 3: Exponential expressions

#### Question

Simplify: $5 2x-1 9 x-2 15 2x-3$

##### Change the bases to prime numbers
$5 2x-1 9 x-2 15 2x-3 =5 2x-1 (3 2 ) x-2 (5×3) 2x-3 =5 2x-1 3 2x-4 5 2x-3 3 2x-3$(3)
##### Subtract the exponents (same base)
$=5 (2x-1)-(2x-3) ×3 (2x-4)-(2x-3) =5 2x-1-2x+3 ×3 2x-4-2x+3 =5 2 ×3 -1$(4)
##### Write the answer as a fraction
$=25 3=81 3$(5)

Important: when working with exponents, all the laws of operation for algebra apply.

### Example 4: Simplifying by taking out a common factor

#### Question

Simplify: $2 t -2 t-2 3×2 t -2 t$

##### Simplify to a form that can be factorised
$2 t -2 t-2 3×2 t -2 t =2 t -(2 t ×2 -2 ) 3×2 t -2 t$(6)
##### Take out a common factor
$2 t -2 t-2 3×2 t -2 t =2 t (1-2 -2 ) 2 t (3-1)$(7)
##### Cancel the common factor and simplify
$2 t -2 t-2 3.2 t -2 t =1-1 4 2=3 4 2=3 8$(8)

### Example 5: Simplifying using difference of two squares

#### Question

Simplify: $9 x -1 3 x +1$

##### Change the bases to prime numbers
$9 x -1 3 x +1=(3 2 ) x -1 3 x +1=(3 x ) 2 -1 3 x +1$(9)
##### Factorise using the difference of squares
$9 x -1 3 x +1=(3 x -1)(3 x +1) 3 x +1$(10)
##### Simplify
$9 x -1 3 x +1=3 x -1$(11)

### Exercise 1:

Simplify without using a calculator:

1. $16 0$

2. $16a 0$

3. $2 -2 3 2$

4. $5 2 -3$

5. $2 3 -3$

6. $x 2 x 3t+1$

7. $3×3 2a ×3 2$

8. $a 3x a x$

9. $32p 2 4p 8$

10. $(2t 4 ) 3$

11. $(3 n+3 ) 2$

12. $3 n 9 n-3 27 n-1$

1. ${16}^{0}=1$

2.  $16{a}^{0}=16\left(1\right)=16$

3. ${\frac{{2}^{-2}}{3}}^{2}=\frac{1}{{2}^{2}{\stackrel{.}{3}}^{2}}=\frac{1}{4×9}=\frac{1}{36}$

4. ${\frac{5}{2}}^{-3}=\left(5\right)\left({2}^{3}\right)=\left(5\right)\left(8\right)=40$

5. ${\left(\frac{2}{3}\right)}^{-3}={\frac{{2}^{-3}}{3}}^{-3}={\frac{{3}^{3}}{2}}^{3}=\frac{27}{8}$

6. ${x}^{2.}{x}^{3t+1}={x}^{2.}{x}^{3t}.{x}^{1}={x}^{2+1}.{x}^{3}t={x}^{3}{x}^{3t}$

7. $3×{3}^{2a}×{3}^{2}={3}^{1+2a+2}={3}^{2a+3}$

8. ${\frac{{a}^{3x}}{a}}^{x}={a}^{3x}.{a}^{-}x={a}^{3x-x}={a}^{2x}$

9. $\frac{32{p}^{2}}{4{p}^{8}}=8{p}^{2-8}=8{p}^{-6}={\frac{8}{p}}^{6}$

10. ${\left(2{t}^{4}\right)}^{3}={2}^{3.}{t}^{4.3}=8{t}^{12}$

11. ${\left({3}^{n+3}\right)}^{2}={3}^{2n+6}$

12. ${\frac{{3}^{n}{.9}^{n-3}}{27}}^{n-1}={\frac{{3}^{n}.{\left({3}^{2}\right)}^{n-3}}{{3}^{3}}}^{n-1}={\frac{{3}^{n}{.3}^{2n-6}}{3}}^{3n-3}={3}^{n+2n-6-\left(3n-3\right)}={3}^{3n-6-3n+3}={3}^{-3}={\frac{1}{3}}^{3}=\frac{1}{27}$