**Video Transcript**

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Transcript
**Zero and Negative Exponents**

In a land of two kingdoms, rival **Kings Wallace the 4th** and Frederick the negative 3rd enjoy playing pranks on each other. King Wallace receives a package from his rival, but is it a package or a ** prank** ? It’s a painting…

Oh my! How provocative! In order to maintain ** diplomatic relations** , the king must hang the painting in a prominent position...this is a terribly tricky situation, so the king calls Mr. Magic, the court mathemagician. Mr. Magic knows just what to do. He’ll shrink the ** painting** From his bag of tricks, he pulls out a secret potion...and leaves the rest to magic.

### The fraction

Oh no! The shrinking potion only worked in one dimension – look what happened. Mr. Magic realizes his **error** ...so he pulls out another **potion** - this time to shrink the provocative painting proportionally by 10⁻⁵. 10⁻⁵! Wait – **negative powers** can be confusing. Let’s **investigate**. Take a look at our problem here, 10⁻⁵. We can rewrite this as a **fraction**. In the denominator, write the base to the absolute value of the power, or 10⁵. So, what do you write in the numerator? 1.

### Denominator and numerator

Now, simplify the fraction. See what happens when you have a positive exponent in the **denominator** of a fraction? The value of the fraction get smaller and smaller. 10⁻⁵ = 1/100,000. I think Mr. Magic is on to something here…Take a look at this **example**: 2⁻⁴. To rewrite this as a fraction, in the denominator, write 2⁴, then, write a one in the numerator, and simplify. 2⁻⁴ = 1/16.

### Example

Let's look at an example when the base is a **variable**. We can rewrite x⁻⁴ as a fraction by writing x⁴, which is x times x times x times x in the denominator and a 1 in the **numerator**. This simplifies to 1/x⁴. Here's the rule for **negative exponents**: x⁻ª = 1/xª. Remember, 'x' cannot be equal to zero. Let's look at a rule to see why this works, and then it will be easier to remember…

### The rule

Our **rule** is: any base raised to the zero power is equal to 1, for example, 1⁰ = 1...2⁰ = 1, and 3⁰ = 1, and so on...the rule is: any base, such as x, raised to the **zero power** is equal to 1, when 'x' does not equal 0.

So, using the example, 2⁻⁴, rewritten as a fraction, is equal to 1/2⁴ is the same as 2⁰ over 2⁴ and like magic, this is equal to 2⁽⁰⁻⁴⁾, which is 2⁻⁴, so we're right back where we started. That makes it much easier to understand! Sometimes **math** is like magic!

King Wallace hung the picture. But wait, where is it? Ah, there. Take a look at it now… Thanks to Mr. Magic, King Wallace isn’t worrying about the **picture** – he’s trying to figure out what prank to play next on King Frederick…