Fractional Exponents Transcript

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So far, we have been dealing with integer exponents.

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This lesson will show you that exponents can be in term of fractions, and some of the basic ideas behind.

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Let's start. We know that root 4 is equals to 2.

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Now, fractional exponents are not very different from what you have seen here.

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Let me show you why. Consider this term, 4 to the power of 1 over 2

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We can rewrite 4 as 2 to the power of 2. So, now we have bracket 2 to the power of 2, to the power of 1 over 2.

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Now, using the third exponent law, this term becomes, 2 to the power of 2 multiply by 1 over 2.

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Now, both of the two cancel off, and we are left with 2.

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Notice that, these 2 numbers are the same.

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Therefore, we can say that, root 4 is equals to 4 to the power of 1 over 2.

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Most importantly, we can conclude that, the root sign is equivalent to this fractional exponent, 1 over 2.

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Let's build up from here. If we change 4 to 'a', root 'a' will be equals to 'a' to the power of 1 over 2.

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If the root is changed to cube root, the exponent will change to, 1 over 3. Do you see the pattern here?

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Finally, if we change 3 to m, the exponent will change to 1 over m.

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So, in general, the m root of 'a' is equals to 'a' to the power of 1 over 'm'.

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Let's analyze this formula further. If 'a' is changed to 'a' to the power of n, we will get bracket 'a' to the power of 1 over m.

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Again, using this exponent law, this term becomes 'a' to the power of 'n' multiply by 1 over 'm'. Hence, we now have 'A' to the power of n over m.

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From here, we can see that, 'a' to the power of n over m is equals to, 'm' root of 'a' to the power of n.

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Let's rewrite these terms here.

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Let's continue, notice that we can switch the two terms here to get, 'A' to the power of 1 over m multiply by n.

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Now, notice that we have 'a' to the power of 1 over m. If we look carefully, this part of the term is also equals to m root of 'a'.

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Therefore, this term can be written as, bracket m root to the power of n.

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With this, we can see that, bracket m root 'a' to the power n is equals to, m root a to the power of n.

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Let's write down this observation here.

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Finally, we have the formula, 'a' to the power of n over m, equals to m root 'a' to the power of n, equals to bracket m root 'a' to the power of n.

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Now, to understand this formula better, let's simplify, 8 to the power of 2 over 3.

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Using this formula, 'a' to the power of n over m, equals to bracket m root n to the power of n, we get 8 to the power of 2 over 3 as, bracket 3 root 8 to the power of 2.

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Root 3 of 8 is 2. Finally, bracket 2 to power of 2 gives 4.

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Now, without using this formula, we can also simplify this example using the usual exponent laws. Let me show you how.

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We know that 8 is equals to, 2 to the power of 3. So let's replace 8 with 2 to the power of 3.

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Let's use the third exponent law to further simplify this term. Bracket 2 to the power of 3, to the power of 2 over 3, gives 2 to the power of 3 multiply by 2 over 3.

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When we multiply 3 with 2 over 3, both threes cancels off. This leaves us with 2 to the power of 2.

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Now, 2 to the power of 2 gives 4. This is the same answer as the previous method.

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Personally, I think this way of simplifying is more elegant. But, it's up to you to choose the way that suits you.

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That's all for this lesson on fractional exponents. Try out the practice question to further your understanding.