# How can you find cubes of numbers without using multiplication?

##### 4 Answers

Use logarithms and exponents.

#### Explanation:

Hmmm. Well I suppose you could use addition, logarithms and exponents...

#x^3 = e^(3 ln x) = e^(e^(ln 3 + ln ln x))#

Or:

#x^3 = 10^(3 log x) = 10^(10^(log 3 + log log x))#

Is that what you were looking for?

**Some more explanation**

The cube of a number

#x^3 = x*x*x#

You can convert multiplication into addition using logs and exponents.

We can use any base of logarithm, but the most commonly used ones are natural logarithm

Note that exponentiation relates multiplication and addition like this:

#a^(m+n) = a^m*a^n#

So we can use it in combination with logarithm to do multiplication using addition:

#xy = a^(log_a x + log_a y)#

As an extension of this we find that:

#log_a x^n = n log_a x#

and hence:

#x^n = a^(n log_a x)#

We can express the multiplication

#n log_a x = a^(log_a n + log_a log_a x)#

Hence putting

#x^3 = e^(e^(ln 3 + ln ln x))#

Putting

#x^3 = 10^(10^(log 3 + log log x))#

Alternatively, we could use addition instead of multiplication by

#x^3 = e^(ln x + ln x + ln x)#

#x^3 = 10^(log x + log x + log x)#

Demonstration example using addition

#### Explanation:

Think of multiplication as addition. So explaining by example.

so lets see if we can change

We know that

But

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So by using addition we have demonstrated the

If you are happy to square numbers, add, subtract and divide by

#x^3 = ((x^2+x)/2)^2 - ((x^2-x)/2)^2#

#### Explanation:

Suppose you are not happy with multiplying two arbitrary numbers but:

- You know the squares of numbers.
- You are happy to add, subtract and divide by
#2# .

Note first that:

#(a+b)^2 - (a-b)^2 = (a^2+2ab+b^2) - (a^2-2ab+b^2) = 4ab#

So:

#((a+b)/2)^2 - ((a-b)/2)^2 = ab#

Put

#x^3 = ((x^2+x)/2)^2 - ((x^2-x)/2)^2#

For example:

#4^3 = ((4^2+4)/2)^2 - ((4^2-4)/2)^2#

#color(white)(4^3) = ((16+4)/2)^2 - ((16-4)/2)^2#

#color(white)(4^3) = (20/2)^2 - (12/2)^2#

#color(white)(4^3) = 10^2 - 6^2#

#color(white)(4^3) = 100 - 36#

#color(white)(4^3) = 64#

Here's a way to construct cubes using just addition and subtraction...

#### Explanation:

Here's a way to construct the sequence of cubes of positive integers without using multiplication...

Write down the first four cubes of positive integers in a line with spaces between them (leaving space on the right too)...

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

In the gaps under each pair of numbers, write the difference between them to make a second line...

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

#color(white)(00)7color(white)(000)19color(white)(000)37#

Add a third line consisting of the differences between each pair of numbers in the second line...

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

#color(white)(00)7color(white)(000)19color(white)(000)37#

#color(white)(0000)12color(white)(000)18#

Add a fourth line containing the difference between the pair of numbers in the third line...

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

#color(white)(00)7color(white)(000)19color(white)(000)37#

#color(white)(0000)12color(white)(000)18#

#color(white)(00000000)6#

Extend the fourth line by repeating the number

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

#color(white)(00)7color(white)(000)19color(white)(000)37#

#color(white)(0000)12color(white)(000)18#

#color(white)(00000000)6color(white)(0000)color(red)(6)color(white)(0000)color(red)(6)#

Construct resulting additional numbers for the third line by adding...

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

#color(white)(00)7color(white)(000)19color(white)(000)37#

#color(white)(0000)12color(white)(000)18color(white)(000)color(red)(24)color(white)(000)color(red)(30)#

#color(white)(00000000)6color(white)(0000)color(red)(6)color(white)(0000)color(red)(6)#

Repeat to get additional terms for the second line...

#1color(white)(0000)8color(white)(000)27color(white)(000)64#

#color(white)(00)7color(white)(000)19color(white)(000)37color(white)(000)color(red)(61)color(white)(000)color(red)(91)#

#color(white)(0000)12color(white)(000)18color(white)(000)color(red)(24)color(white)(000)color(red)(30)#

#color(white)(00000000)6color(white)(0000)color(red)(6)color(white)(0000)color(red)(6)#

Finally repeat for the first row to get:

#1color(white)(0000)8color(white)(000)27color(white)(000)64color(white)(00)color(red)(125)color(white)(00)color(red)(216)#

#color(white)(00)7color(white)(000)19color(white)(000)37color(white)(000)color(red)(61)color(white)(000)color(red)(91)#

#color(white)(0000)12color(white)(000)18color(white)(000)color(red)(24)color(white)(000)color(red)(30)#

#color(white)(00000000)6color(white)(0000)color(red)(6)color(white)(0000)color(red)(6)#