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Calculus: Help and Review13 chapters | 148 lessons

Instructor:
*Stephanie Matalone*

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, you will learn the steps of solving 5 to the negative 4th power. In doing so, you will learn the basics of exponents and negative exponents and how to solve them.

You are deciding whether or not to enter a raffle. Your smart friend thinks it will be funny to tell you that you have a 5 to the negative 4th power chance of winning the raffle if you buy a ticket. Should you spend the $5? Is it worth it?

Before you spend your money, you want to know what 5 to the negative 4th power even means! Well, the word power tells you that you are dealing with exponents. In our problem, -4 is the exponent and 5 is the base. When we write this out, the exponent -4 will be the small number written like a superscript after the base of 5, which is written in normal size.

This is how it looks when written:

An exponent can also be written out like this:

5^-4

**Exponents** tell you how many times to multiply the **base** number by itself. Typically, this is pretty easy when you have a positive exponent. But, how do you multiply 5 by itself -4 times? The easy way to do this is by writing a fraction with the base and positive exponent in the **denominator**, or bottom of the fraction. You will then put a 1 in the **numerator**, or top of the fraction.

As a fraction, it will look like this:

The reason why you can put negative exponents in the denominator is because of inverse operations. The expression *x*^0 is is equal to 1 and the expression *x*^1 is equal to *x*. To go from *x*^0 to *x*^1, we just multiply by *x*. The expression *x*^2 is equal to *x* x *x* and *x*^3 is equal to *x* x *x* x *x*. To go from *x*^2 to *x*^3, we again multiply by *x*.

To go backwards from *x*^3 to *x*^2, we will use the inverse operation of multiplication, which is division. Thus, we will divide by *x* which is the same as multiplying by 1/*x*. Let's say we have *x*^-5, we would be dividing *x*^0 or 1 by *x*^5, which is the same as multiplying it by 1/*x*^5. Thus, *x*^-5 is equal to to 1 times 1/*x*^5 which is 1/*x*^5.

Now that you have a positive exponent in the denominator, you will treat it like a normal exponent problem. In the denominator, you will expand the exponent into its true multiplication. Since the exponent is four, we know we need to multiply 5 by itself four times so we will write that out:

Next, we will simply multiply out the 5s in the denominator. It is smart to multiply 2 numbers at a time so you do not make a mistake.

5 to the negative 4th power is equal to 1/625. This means that you have a 1 in 625 chance of winning the raffle! Not bad odds! Probably worth the five dollars on the ticket.

In order to check your work and ensure you really want to enter the contest, you must work backwards. To work backwards with exponents, we must use a root. In this case, we will use the **fourth root**, which means finding out what can be multiplied by itself four times to give you the number.

We are only going to deal with the denominator and make sure we did that work correctly. To do so, we will take the fourth root of 625 (rather than 1 / 625). This means we must see what number can be multiplied by itself 4 times to get 625. The fourth root of 625 is 5 because 5 x 5 x 5 x 5 is equal to 625. Because we got our original base number, it is clear we did our work correctly.

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Calculus: Help and Review13 chapters | 148 lessons

- What is a Function: Basics and Key Terms 7:57
- Graphing Basic Functions 8:01
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
- Polynomial Functions: Properties and Factoring 7:45
- Polynomial Functions: Exponentials and Simplifying 7:45
- Exponentials, Logarithms & the Natural Log 8:36
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- Equation of a Line Using Point-Slope Formula 9:27
- Horizontal and Vertical Asymptotes 7:47
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- Normal Line: Definition & Equation
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- Point Slope Form: Definition, Equation & Example
- Point Symmetry: Definition & Examples 3:09
- Polar Coordinates: Definition, Equation & Examples
- Negative Reciprocal: Definition & Examples 3:39
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- Solving 5 to the Negative 4th Power
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