Hello. In this math educational post, I go over working with negative exponents.

Rendered math text is from the use of QuickLaTeX.com

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## Topics
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* Exponents Review
* Dealing With Negative Exponents
* Practice Problems
* Answers To Practice Problems

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## Exponents Review
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An exponent can be viewed as a number counter of how many times a number is multiplied by itself. Instead of writing 2 x 2 x 2 x 2 x 2, we can write https://quicklatex.com/cache3/ed/ql_b743a7a19ccbf511aba5d3dc565942ed_l3.png where the base 2 is the number being multiplied by itself. The exponent 5 represents how many times the two is used when multiplying by itself. 

When it comes to fractional exponents it is kind of difficult to explain what it is. Having https://quicklatex.com/cache3/f4/ql_ec930800d1e902aafa1a6b05db2b36f4_l3.png is actually taking the square root of 2. When it comes to https://quicklatex.com/cache3/e6/ql_63249a44309c53b032bb55db1b491ee6_l3.png, this is taking the fifth root of 7. Consider the case where the exponent does not have a numerator of one such as https://quicklatex.com/cache3/35/ql_f5d93f3f318255c4622d8c2360218635_l3.png. With https://quicklatex.com/cache3/35/ql_f5d93f3f318255c4622d8c2360218635_l3.png I do ten squared to obtain 100 and then take the third root of 100. 

With a non-zero base and an exponent zero the number results in a 1. Examples include https://quicklatex.com/cache3/10/ql_1394f90b41784e6aa1cd2ea4b57e2b10_l3.png in which they equal one.

<center><img src="https://cdn.pixabay.com/photo/2020/08/03/05/03/students-5459157_960_720.jpg" /></center>
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## Dealing With Negative Exponents
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Negative exponents operate differently than positive exponents. It is common math practice to convert negative exponent values into its equivalent value with positive exponents.

Given https://quicklatex.com/cache3/40/ql_f19ae5d87903b040c29110a52a5e2840_l3.png this turns into its reciprocal of 1 divided by x:

<center>https://quicklatex.com/cache3/40/ql_173839164bb590969a9bc54809056e40_l3.png</center>

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The more confusing variation of the above is dealing with https://quicklatex.com/cache3/23/ql_6c1cc17da6201bc59e67fe8dc9bf0f23_l3.png which ends up being x.

<center>https://quicklatex.com/cache3/f8/ql_dc594655fb9f04fac6d1d6334e9caef8_l3.png</center>

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**Example One**

Evaluate https://quicklatex.com/cache3/1f/ql_7d0c51b15e1c7087a564ec1817ea051f_l3.png.

**Solution**

<center>https://quicklatex.com/cache3/1d/ql_b9234e31c1173a356c582d367bc0721d_l3.png</center>

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**Example Two**

Evaluate one tenth to the power of negative 2.

<center>https://quicklatex.com/cache3/49/ql_999c17f1c991ad7a3336c4d945499a49_l3.png</center>

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**Example Three**

Convert the following expression into positive exponents.

<center>https://quicklatex.com/cache3/9b/ql_a9bd4b5a4edf27ac86fdc45c8a4cc89b_l3.png</center>

**Solution**

<center>https://quicklatex.com/cache3/77/ql_aae990cd5f4e0ace6da05878bfa02677_l3.png</center>

With this case, you can "flip" the fraction and remove the negative signs from the exponents.

<center><img src="https://cdn.pixabay.com/photo/2013/05/16/09/05/mathematics-111423_960_720.jpg" /></center>
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## Practice Problems
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Evaluate the following into a single number.

1) https://quicklatex.com/cache3/1f/ql_7d0c51b15e1c7087a564ec1817ea051f_l3.png

2) https://quicklatex.com/cache3/76/ql_c2948cfcb1a5dac884e01b355af51776_l3.png

3) https://quicklatex.com/cache3/e1/ql_1e12a6c385076ada25da4f1c9693b6e1_l3.png

4) https://quicklatex.com/cache3/ec/ql_8b19a11878541948a99d2f62f70e24ec_l3.png

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Convert into positive exponents.

5) https://quicklatex.com/cache3/02/ql_938e121c89001a3e178a9f766c628802_l3.png

6) https://quicklatex.com/cache3/62/ql_8b3c6be3b430163addd5b69088bec562_l3.png

7) https://quicklatex.com/cache3/2b/ql_fd31e13455d68ced2a11382c07a58b2b_l3.png

8) https://quicklatex.com/cache3/c6/ql_3de18c00711f50992bcb8f49c245b9c6_l3.png


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## Answers To Practice Problems
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1) 1/8

2) 1/9

3) 1/4

4) 49/4

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Convert into positive exponents.

1) https://quicklatex.com/cache3/87/ql_b4349f6aa682c50f27b4d142a0834b87_l3.png

2) https://quicklatex.com/cache3/f5/ql_9337ef8c0c4356c8ea9c8b8df36b6bf5_l3.png

3) https://quicklatex.com/cache3/0b/ql_31576f7a1967b53395d190fdc5feb80b_l3.png

4) https://quicklatex.com/cache3/c5/ql_4d20ba16c66e033f2f38c0057bce09c5_l3.png

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<center><img src="https://cdn.pixabay.com/photo/2015/11/15/07/43/mathematics-1044075_960_720.jpg" /></center>
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<center>Thank you for reading.</center>