Math Is Fun Forum

Jordan123

Guest

Math is frustrating

How do square roots and exponents work????

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Math is frustrating

hi Jordan123

Welcome to the forum.

I think you're wondering why

Power notation was invented as a shortcut for repeated multiplication

From this we have the three laws for powers:

For all of this it is assumed that m and n are positive integers and they're easy to prove.

But mathematicians don't like limitations like this (the positive integer bit) if it can be avoided. So they have sought to extend the definition whilst maintaining the three laws.

So let's start with "What could a zero power mean?"

From law 2, if m = n we have

But a^m over a^m must be 1, so let's define a^0 to be 1 for all values of a.

Now for a fraction like 1/2

From law 1

So a to the power 1/2 behaves exactly like the square root of a, so let's define it to be that.

There's loads more of this sort of thing, all created definitions based on the three laws.  It leads to a whole area of maths that is consistent and useful so it has kind of stuck, And why not?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Math is frustrating

hi Jordan123

Welcome to the forum.

I think you're wondering why

Power notation was invented as a shortcut for repeated multiplication

From this we have the three laws for powers:

For all of this it is assumed that m and n are positive integers and they're easy to prove.

But mathematicians don't like limitations like this (the positive integer bit) if it can be avoided. So they have sought to extend the definition whilst maintaining the three laws.

So let's start with "What could a zero power mean?"

From law 2, if m = n we have

But a^m over a^m must be 1, so let's define a^0 to be 1 for all values of a.

Now for a fraction like 1/2

From law 1

So a to the power 1/2 behaves exactly like the square root of a, so let's define it to be that.

There's loads more of this sort of thing, all created definitions based on the three laws.  It leads to a whole area of maths that is consistent and useful so it has kind of stuck, And why not?

Bob

Excellent notes. How about a^(3/2)?

This is sqrt{a^3}.

Yes?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Math is frustrating

Yes, from the 3rd law:

or alternatively

You can check this out with an example.

Let a = 64

(root 64)^3 = 8^3 = 512

root (64^3) = root(262144) = 512

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Math is frustrating

Yes, from the 3rd law:

or alternatively

You can check this out with an example.

Let a = 64

(root 64)^3 = 8^3 = 512

root (64^3) = root(262144) = 512

Bob

Honestly, math is simply never-ending education.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,837

Re: Math is frustrating

Hi,

The Laws of Exponents and Radicals explained in these links:

Exponents Laws

Fractional Exponents.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Math is frustrating

Hi,

The Laws of Exponents and Radicals explained in these links:

Exponents Laws

Fractional Exponents.

Yes, I know that the site has links to math notes. The idea for posting threads is to get people involved in mathematical discussion.