1. Writing Radicals in Rational Form

2. DEFINITIONS EXPONENT BASE
Base: The term/variable of which is being raised upon
Exponent: The term/variable is raised by a term. AKA Power
EXPONENT
BASE

3. THE EXPONENT

4. 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚 NTH ROOT RULE m is the power (exponent)
n is the root or index
a is the base or once in radical form a is the radicand
𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚

5. RULES Another way of writing 𝟐𝟓 is 𝟐𝟓 𝟏 𝟐 .
𝟐𝟓 is written in radical expression form.
𝟐𝟓 𝟏 𝟐 is written in rational exponent form.

6. Evaluate 43/2 in radical form and simplify.
EXAMPLE 1
Evaluate 43/2 in radical form and simplify.
𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚

7. EXAMPLE 2
Evaluate 𝟒 𝟏 𝟐 in radical form and simplify. 4

8. YOUR TURN Evaluate −𝟐𝟕 𝟐 𝟑 in radical form and simplify. 3 −27 2 =
3 − =
(−3) 2 = 9

9. 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚 EXAMPLE 3 Evaluate −𝟐𝟕 𝟒 𝟑 in radical form
and simplify.
𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 or 𝑛 𝑎 𝑚
Use calculator to check
Hint: Remember, the negative is OUTSIDE of the base

10. EXAMPLE 4
Evaluate in radical form and simplify.
or 𝑥−1 3
5 4𝑥 −1 3

11. NTH ROOT RULE
m is the power (exponent)
n is the root
a is the base

12. EXAMPLE 5
Evaluate (27)–2/3 in radical form and simplify.

13. EXAMPLE 6
Evaluate −𝟔𝟒 − 𝟐 𝟑 in radical form and simplify.
3 −64 −2 =
−4 −2 =
1 − =
1 16

14. YOUR TURN
Evaluate 𝟐𝟓 𝟑𝟔 − 𝟏 𝟐 in radical form and simplify.
25 − −1 = =
6 5

15. PROPERTIES OF EXPONENTS
Product of a Power:
Power of a Power:
Power of a Product:
Negative Power Property:
Quotient Power Property:

16. If the BASES are the same, ADD the powers
EXAMPLE 7
Simplify
Saying goes: BASE, BASE, ADD
If the BASES are the same, ADD the powers

17. EXAMPLE 8
Simplify
𝑥 • 𝑥 =
𝑥 =
𝑥 =
𝑥 5 6

18. YOUR TURN
Simplify
𝑥 • 𝑥 =
𝑥 =
𝑥 = 𝑥

19. EXAMPLE 9 Simplify Saying goes: POWER, POWER, MULTIPLY
If the POWERS are near each other, MULTIPLY the powers – usually deals with PARENTHESES

20. EXAMPLE 10
Simplify
2 𝑥 = 𝑥

21. YOUR TURN 3 𝑥 1 4 𝑦 − 2 3 4 = 3 4 𝑥 1 4 4 𝑦 −2 3 4 = 81𝑥 𝑦 −8 3 =
Simplify
3 𝑥 𝑦 − = 𝑥 𝑦 −
= 81𝑥 𝑦 − =
81𝑥 𝑦 8 3

22. EXAMPLE 11 = 7 − 3 3 = 7 −1 = Simplify
Saying goes: When dividing an expression with a power, SUBTRACT the powers.
= 7 − 3 3
= 7 −1 =

23. EXAMPLE 12
Simplify
𝑥 𝑥 =
𝑥 − =
𝑥 − =
𝑥 1 6

24. EXAMPLE 13
Simplify
𝑥 • 𝑥 1 4
𝑥 𝑥 1 5 𝑥 𝑥
𝑥 − 1 5
𝑥 7 12
𝑥 − 3 15
𝑥 •𝑥 7 12
𝑥 2 15
𝑥 =
𝑥 = 𝑥