If we want to multiply two powers with the same base number, e.g. 4³ * 4⁵ we do the following:

4³ = 4 * 4 * 4  

and  

4⁵ = 4 * 4 * 4 * 4 * 4,

so

4³ * 4⁵ = (4 * 4 * 4) * (4 * 4 * 4 * 4 * 4) = 4⁸ = 4³⁺⁵

In general:

The product of two powers with the same base number is the same base number whose index is the sum of the other two indices.

 a^m * aⁿ = a^m+n

9. Work out the following in index form and write your answers in your notebook:

a) 2³ * 2⁷    b) 3⁵ * 3³ ;   c) 5⁵ * 5³

d) 2^-3 * 2⁵    e) 3^-5 * 3^-3 ;   f) 5^-5 * 5³

Check your results in the following window.

Index 1 is the exponent of the first factor; Index 2 is the exponent of the second factor. Increase the number of decimal places if necessary.

10. Work out the following and write the answers in your notebook in index form: 

a) 2 * 2⁴ * 2⁵    b) 4² * 4⁴ * 4³
c) 8 * 8 * 8⁴

d) 2 * 2^-4 * 2⁵    e) 4^-2 * 4⁴ * 4^-3
f) 8^-1 * 8 * 8⁴

Check your results in the following window.

Index 1 is the exponent of the first factor; Index 2 is the exponent of the second factor and Index 3 is the exponent of the third.

4⁵ : 4³ = (4 * 4 * 4 * 4 * 4) : (4 * 4 * 4) = 4² = 4^5-3

11. Calculate the following and write the answers in index form in your notebook:

a) 2⁷ : 2³    b) 3⁵ : 3³    c) 5⁶ : 5³

d) 2⁷ : 2^-3    e) 3^-2 : 3²    f) 5^-4 : 5^-3

Check your results in the following window.

Index 1 is the exponent of the numerator; Index 2 is the exponent of the denominator.

To work out (2*3)²  we have to do the following:

(2*3)³ = (2*3) * (2*3) * (2*3) = (2*2*2) * (3*3*3) = 2³ * 3³

To calculate the result we have to multiply 2*3 and cube the product: (2*3)³ = 6³ = 216

Or, we can cube each of the factors, 2³ = 8  and  3³= 27, and multiply the result: 8*27 = 216.

In general:

A product raised to a power is equal to multiplying these numbers raised to the same power

(a*b)m = am * bm

12. Express the following in product form:

a) (2*5)⁶    b) (3*4)²

c) (2*8)³    d) (4*6)⁴

e) (2*5)^-2    f) (3*2)^-3    g) (2*5)^-3

Work out the answers in your notebook and check them in the following window.

Likewise, we can easily deduce that:

(a/b)^m = a^m / b^m

13. Express the following in division form:

a) (18/2)⁶    b) (8/4)²

c) (10/5)³    d) (12/3)⁴

e) (18/2)^-3    f) (8/4)^-2

g) (10/5)^-3    h) (9/3)^-4

Work out the answers in your notebook and check them in the following window. 

If we want to work out (4⁵)³ we have to do the following:

(4⁵)³ = 4⁵ * 4⁵ * 4⁵ = 4⁵⁺⁵⁺⁵ = 4^5*3

Therefore we can deduce the following rule:

A power raised to another power is the same as the base number raised to the product of these two powers:

(a^m)ⁿ = a^m*n

14. Simplify the following in your notebook and express each example as a number raised to one power:

a) (2³)⁷    b) (3⁵)³    c) (5⁵)³

d) (2^-3)²    e) (3³)^-2    f) (5^-2)^-3

Work out the answers in your notebook and check them in the following window.