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Evaluating radicals and integer exponents

Author: Jesus Gonzalez

What we know

We know how to deal with positive whole number exponents

3 to the power of 4 cross times 3 to the power of 2 equals end exponent
left parenthesis 3 cross times 3 cross times 3 cross times 3 right parenthesis cross times left parenthesis 3 cross times 3 right parenthesis equals
3 cross times 3 cross times 3 cross times 3 cross times 3 cross times 3 equals
3 to the power of 6

Because the two terms shared the same base, we were able to combine them through multiplication. The net result is that multiplying the whole terms leads to adding the exponents.

$3 to the power of 4 cross times 3 squared space equals space 3 to the power of open parentheses 4 plus 2 close parentheses end exponent space equals space 3 to the power of 6$

Now let's consider negative exponents

Source: https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals/pre-algebra-negative-exponents/v/negative-exponent-intuition

Negative Exponents

We can apply what we already knew about positive exponents and apply it to negative exponents.

$4 to the power of 5 cross times 4 to the power of negative 3 end exponent$ should equal $4 to the power of 5 plus left parenthesis negative 3 right parenthesis end exponent$ since we noticed before that exponents can just be added together when they have the same base.

And we should get $4 squared$

Notice that $4 to the power of 5 cross times 1 over 4 cubed equals 4 squared$ as well from what we knew before. So the key idea to take away here is that $4 to the power of negative 3 end exponent equals 1 over 4 cubed$

Or more generally: $a to the power of negative b end exponent equals 1 over a to the power of b$ where a and b are real numbers.

What about rational exponents?

Source: https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/rational-exponents-intro/v/basic-fractional-exponents

Rational Exponents

The rule we've been relying on is that when multiplying two term, if the bases match you can add the exponents to combine those terms.

Well now we have to consider something like $3 to the power of 1 half end exponent$ or $5 to the power of 1 third end exponent$

Since $1 half plus 1 half equals 1$ , we would like $3 to the power of 1 half end exponent cross times 3 to the power of 1 half end exponent$ to equal $3 to the power of 1$ , since that would agree with the pattern we've noticed so far. This is very similar to what we've learned about roots.

$square root of 3 cross times square root of 3 equals 3$

So the general rule to pull from this is that $square root of 3 equals 3 to the power of 1 half end exponent$ , or more generally: $a to the power of 1 over b end exponent equals b-th root of a$ , where a and b are real numbers.

note: When working with numerator that are not 1, you can apply the rules we've learned about exponents on exponents, to simplify something like $5 to the power of 3 over 4 end exponent$ into $5 to the power of open parentheses 1 fourth x 3 over 1 close parentheses end exponent$ or $open parentheses 5 to the power of 1 fourth end exponent close parentheses cubed$

Now you're ready for the quiz

Take some time now and consider these 4 questions. Feel free to review what we've discussed.

Once you're done it's time for the Exit Ticket: Here