Laws Of Exponents
Laws of Exponents
Exponents are also called Powers or Indices
The exponent of a number says how many times to use the number in a multiplication.
In this example: 82 = 8 × 8 = 64
In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared" .
So an Exponent just saves you writing out lots of multiplies!
Example: a7 a7 = a × a × a × a × a × a × a = aaaaaaa
Notice how I just wrote the letters together to mean multiply? We will do that a lot here.
Example: x6 = xxxxxx The Key to the Laws
Writing all the letters down is the key to understanding the Laws
Example: x2x3 = (xx)(xxx) = xxxxx = x5
Which shows that x2x3 = x5, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.
All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:
The exponent says how many times to use the number in a multiplication.
A negative exponent means divide, because the opposite of multiplying is dividing
A fractional exponent like 1/n means to take the nth root:
If you understand those, then you understand exponents!
And all the laws below are based on those ideas.
Laws of Exponents
Here are the Laws (explanations follow):
Law
Example x1 = x
61 = 6 x0 = 1
70 = 1 x-1 = 1/x
4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5 xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn
(x2)3 = x2×3 = x6
(xy)n = xnyn
(xy)3 = x3y3
(x/y)n = xn/yn
(x/y)2 = x2 / y2 x-n = 1/xn x-3 = 1/x3
And the law about Fractional Exponents:
Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:
Example: Powers of 5 .. etc..
52
1 × 5 × 5
25
51
1 × 5
5
50
1
1
5-1
1 ÷ 5
0.2
5-2
1 ÷ 5 ÷ 5
0.04
.. etc..
Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5
Exponents are also called Powers or Indices
The exponent of a number says how many times to use the number in a multiplication.
In this example: 82 = 8 × 8 = 64
In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared" .
So an Exponent just saves you writing out lots of multiplies!
Example: a7 a7 = a × a × a × a × a × a × a = aaaaaaa
Notice how I just wrote the letters together to mean multiply? We will do that a lot here.
Example: x6 = xxxxxx The Key to the Laws
Writing all the letters down is the key to understanding the Laws
Example: x2x3 = (xx)(xxx) = xxxxx = x5
Which shows that x2x3 = x5, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.
All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:
The exponent says how many times to use the number in a multiplication.
A negative exponent means divide, because the opposite of multiplying is dividing
A fractional exponent like 1/n means to take the nth root:
If you understand those, then you understand exponents!
And all the laws below are based on those ideas.
Laws of Exponents
Here are the Laws (explanations follow):
Law
Example x1 = x
61 = 6 x0 = 1
70 = 1 x-1 = 1/x
4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5 xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn
(x2)3 = x2×3 = x6
(xy)n = xnyn
(xy)3 = x3y3
(x/y)n = xn/yn
(x/y)2 = x2 / y2 x-n = 1/xn x-3 = 1/x3
And the law about Fractional Exponents:
Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:
Example: Powers of 5 .. etc..
52
1 × 5 × 5
25
51
1 × 5
5
50
1
1
5-1
1 ÷ 5
0.2
5-2
1 ÷ 5 ÷ 5
0.04
.. etc..
Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5