Today we will be discussing the properties of exponents.
The properties of exponents allow us to simplify expressions involving large amounts of exponents.
In order to use the properties of exponents, our bases must be the same. To illustrate this idea, let’s introduce the first of our properties.
Our first property is called the multiplication rule.
When we multiply two exponents of the same base, we can simplify the expression by adding together the exponents and keeping the base the same.
For example if we want to find the result of multiplying 213 and 221, the result would be such a large number, that it makes more sense to use our multiplication rule rather than trying to find the value of each of these numbers.
Since our bases are both 2, we can add together the exponents to get our simplified result.
213 × 221 = 213 + 21 = 234
If we were asked to multiply 213 by 321, we would not be able to use the multiplication rule since our bases are not the same.
In this case, our base is 2 and in this our base is 3.
The next property of exponents is the division rule. When we divide two exponents of the same base, we can simplify the expression by subtracting the exponent of the number at the bottom of our fraction.
Let’s find the result of dividing x4 by x7. Since both our bases are the same, we can use the division rule and subtract our exponents to get the simplified result.
In this case, our result is x−3.
So what does it mean to have a negative exponent above our base?
is the same expression as .
Our third property can be illustrated with the following equation:
The next property of exponents is the distributive rule. The distributive rule allows us to distribute an exponent across a set of parentheses.
In order to distribute an exponent across each term in a set of parentheses, we must multiply each exponent by the exponent on the outside of the parentheses.
Let’s simplify the following expression.
To simplify this expression, we should multiply the exponent of each term by the exponent on the outside.
This expression will become 53x12y6. The coefficient 5 has an invisible exponent of 1, so we need to make sure we also multiply the exponent by 3.
Occasionally, we may have an exponent that is presented as a fraction. Fractional exponents mean that we are dealing with the radical symbol.
In general, we can use the following formula to convert between fractional exponents and radicals.
For example, 4⅔ is equal to the third root of 42.
We can also convert from a radical to a fractional exponent.
For example, the fifth root of 125 is also equal to 125⅕.
We may be asked questions which involve many of the properties we discussed. As long as we apply the correct rules, we will always simplify our expressions correctly.
Let’s simplify the following expression:
We can simplify the numerator of this fraction by adding our exponents together. Our result will be:
From here, we can subtract our two exponents and get the result of:
Let’s take a look at another problem. We want to simplify this expression using the distributive rule.
To simplify this expression, we need to multiply the exponents of each of our terms by 2. Our result will be:
52x10y4 which is also equal to 25x10y4
Let’s take a look at one more problem.
Here we have two fractional exponents that are being divided by each other. Since our bases are the same, we can subtract the exponents from each other and we will get the result of:
Since this is a fractional exponent, it will also be equal to the second root of x, or the square root of x.
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