REVIEW OVER RADICALS/ROOTS

OBJECTIVE FOR THE DAY

• Go over roots
• Review over some rules for roots
• Simplify roots
• Examples
• Now try some on your own!

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So, then what are roots?

A root is our way to undo an exponent.

An exponent tells you how many times you need to multiply a number to get another number.

For example:

Is really just a shorter way of writing:

Now, a root is the opposite of that.

A root tells us how many times a number needs to be multiplied to become the number on the inside.

This sounds really complicated, but let’s use our example.

We know 4 to the 4^th power is 256.

So let’s do the opposite, or take the 4^th root of 256:

What this is asking is:

Which we know is actually:

But, since we only need one of them as an answer, our answer would be:

The rule for even square roots

So, technically what we just got is true.

However, an unwritten rule for even roots is to also include its negative component.

So, for our last example:

Even though we just proved that it is equal to:

It’s also equal to:

But that makes sense right?

If you’re not sure, let’s try it out.

So, as we can see, -4 is also a fourth root of 256.

Explanation of roots importance

So, roots are important because they are the inverse operation to exponential expressions.

Remember, an inverse operation is the opposite operation to another operation.

An example of this is subtraction.

The opposite of subtraction is….

Addition.

So they are inverse operations.

Same with multiplication. The inverse to multiplication is…..

Division.

So what’s the inverse operation to exponential expressions?

Roots.

SO WHAT DOES THIS ALL MEAN?

9,-9 = x

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So what did we just do there?

We took the root of each side of the equation (because what you do to one side, you need to do to another).

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A FEW RULES TO CONSIDER

Multiplying roots

You can multiply roots together, but only multiply.

An example of this is:

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We know this works because we can also just solve the root problems and then multiply.

So:

You can also add roots together, however only if they are in the root.

Example:

Which is not the same as:

You can also divide roots together if you need to.

Example:

We know this works because we can also just solve the root problems and then divide.

So:

= 9

= 9

= 10, -10

= 6 + 8

= 14

YOU CAN ALSO BREAK SQUARE ROOTS

SIMPLIFIED PROPERLY

It’s also proper to not leave radicals in the denominator of a fraction.

An example is:

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(Since having the same number in the numerator and denominator is actually 1)

So we have:

Example 1

So now we divide the equation by x and we get:

x = -9,9

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So now we can solve exponential equations much easier.

EXAMPLE 2

EXAMPLE 3

Here’s another example.

Let’s say we have something like:

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________

3 3

Example 4

Last example.

Let’s say we have something like:

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__________

5 5