### Exponent Law - Part 2

#### Lesson Objective

This lesson shows you the basics behind the next three exponent laws. You will learn how these laws are derived...

#### About This Lesson

This lesson is a continuation from the Exponent Laws - Part 1 lesson.

The next three laws that you are going to learn can be derived from the first three laws shown in Part 1. So, the ideas behind next three laws are not completely new.

Again, it is important for you to understand these laws before using them. This helps to minimize mistakes.

### Study Tips

#### Tip #1

Let's recall the first three exponent laws before proceeding further. The picture below shows the 1st law. Here is an example on how to use it:

6a^{10} x 3a^{-5} = 6 x 3 x a^{10} x a^{-5}

= 18 x a^{10 + (-5)}

= 18 x a^{10 -5 }

= 18a^{5}

#### Tip #2

As for the 2^{nd} law of exponent, the picture below shows the law. Here is an example on how to use it:

6a^{10} ÷ 3a^{-5} = 2a^{10} ÷ a^{-5}

= 2a^{10 - (-5)}

= 2a^{10 + 5}

= 2a^{15}

#### Tip #3

Finally, for the 3^{rd} law of exponent, the picture below shows the law. Here is an example on how to use it:

2(a^{10})^{-5} = 2 x a^{10 x (-5)}

= 2a^{-50}

#### Tip #4

You may come across terms like a^{1} or 2^{1}. These terms simply mean that:

a^{1} = a

2^{1} = 2

We can explain them using the following observation:

2^{3} = 2 x 2 x 2

2^{2} = 2 x 2

Therefore: 2^{1} = 2

Now, watch the following math video to learn the next three laws.

### Math Video

#### Click play to watch

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#### Math Video Transcript

00:00:01.060

This lesson is a continuation from part 1 of the Exponent Laws.

00:00:07.020

To emphasize, it is very important to understand the logic behind these laws before you use them.

00:00:13.110

The laws that you are going to learn can be derived from the first three laws in part 1. So, they are not completely new.

00:00:22.050

Let's start, consider this term, bracket 2 multiply by 3 , to the power of 4.

00:00:30.060

Now, we can break this term into four parts of bracket 2 multiply by 3.

00:00:36.220

We can see that 2 multiply by 2, multiply by 2, multiply by 2, gives 2 power of 4.

00:00:45,070

Similarly, 3 multiply by 3, multiply by 3, multiply by 3, gives 3 power of 4

00:00:54.110

Now, notice this, this term bracket 2 multiply by 3 to the power of 4 is equals to, 2 to the power of 4 multiply by 3 to the power of 4.

00:01:04.210

With this observation, we can now formulate an exponent law.

00:01:09.160

Let's first replace these numbers with alphabetical letters. This gives bracket 'a' multiply by 'b' to the power of n.

00:01:19.140

By referring to this, we can also see this term, bracket 'a' multiply by 'b' to the power of N is equals to, 'a' to the power of 'n' multiply by 'b' to the power of n.

00:01:33.100

So, we have another law of exponent. Bracket 'a' multiply 'b' to the power of 'n' is equals to, 'a' to the power of 'n' multiply b' to the power of 'n'.

00:01:46.060

Let's take a look at the next law of exponent. For us to understand this law, let's consider dividing 2 to the power of 3 with 2 to the power of 3.

00:01:56.200

Now, using the second exponent law, we can see that, this term is equals to, 2 to the power of 3 minus 3.

00:02:08.090

Three minus by three is 0. So, now we have 2 to the power of 0.

00:02:14.240

Alright, let's calculate this term using another way. We can see that 2 to the power of 3 divides by 2 to the power of 3 is equals to, 8 divides by 8.

00:02:26.200

Now, 8 divides by 8 gives 1.

00:02:30.220

From here, we can see something interesting. Since all these terms are equal to each other, we can say that, 2 to the power of zero is equals to 1.

00:02:41.230

With this observation, we can now formulate an exponent law.

00:02:46.200

Let's first replace this number, '2' with 'a'. This gives 'a' to the power of zero.

00:02:54.160

By referring to this, we can deduce that, 'a' to the power of zero is equal to 1.

00:03:02.000

So, here we have another law of exponents. 'a' to the power of zero equals to 1.

00:03:02.000

Let's take a look at the next law of exponent. For us to understand this law, let's consider this term, 1 divides by 2 to the power of 3.

00:03:20.030

Now, notice the number 1, at the numerator. Using this exponent law, 'a' to the power of zero equals to 1, we can rewrite 1 as, 2 to the power of 0.

00:03:36.120

Now, using the second law of exponents, 2 to the power of 0 divides by two to the power of 3 is equals to, 2 to the power of 0 minus 3.

00:03:50.100

0 minus 3 gives negative 3. So now we have, 2 to the power of negative 3.

00:03:58.150

Again, we can see something interesting. If we observe carefully, since all these terms are equal, we can see that 2 to the power of negative 3 is equals to 1 divides by 2 to the power of 3.

00:04:12.090

Let's rewrite these two terms here.

00:04:15.240

With this observation, we can now formulate an exponent law.

00:04:20.140

Let's first replace these numbers with alphabetical letters. This gives 'a' to the power of negative 'n'.

00:04:29.160

By referring to this, we can deduce that, 'a' to the power of negative 'n' is equals to, 1 divides by 'a' to the power of 'n'.

00:04:41.080

Finally, we have another law of exponents. 'a' to the power of negative 'n' equals 1 divides 'a' to the power of 'n'.

00:04:50.160

That is all for this lesson. The next lesson will show you some examples on using what you have learn in this lesson.

### Practice Questions & More

#### Multiple Choice Questions (MCQ)

Now, let's try some MCQ questions to understand this lesson better.

You can start by going through the series of questions on exponent laws - Part 2 or pick your choice of question below.

- Question 1 on the basics of the next three exponent laws

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