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Roots & Exponents
BEDMAS
SQUARE ROOTS
Perfect Squares
Square Roots
Example) Show that 49 is a square number.
Step 1) Draw a square with area 49 square units
Step 2) Determine that each side length of the square is 7 units.
Then, 49 = 7 x 7 = 7
We say: Forty-nine is seven squared
2
area = 49m
2
7m
7m
Example 2) Find the square of 5.
The square of 5 is: 5 = 5 x 5
2
= 25
Example 5) Find the square root of 25.
The square root of 25 is:
√25
Step 1) Find a factor of 25 that occurs twice or when multiplied by itself produces 25.
area = 25m
2
5m
5m
Example 3) Find the square of 6.
The square of 6 is: 6 = 6 x 6
2
= 36
Example 4) Find the square of 7.
The square of 7 is: 7 = 7 x 7
2
= 49
Drawing the square and finding its side length is another method to solve.
5 x 5 = 25
So, we say the square root of 25 is 5.
EXPONENTS
What is an Exponent?
Example Power:
5 x 5 x 5 is 5
5 is the power
5 is the base
3 is the exponent
5 is a power of 5
We say: 5 to the 3rd power, or 5 cubed
3
3
5
3
Power
Exponent
Base
Powers of 10 & The Zero Exponent Law
The Power Laws
Law I
Law II
Law III
Law for Products of powers
Law for Quotients of Powers
Law for Power of Powers
We add the exponents when we multiply powers with the same base:
We subtract the exponents when we divide powers with the same base:
We multiply the exponents when we have a power with a powers with the same base:
2 x 2 = (2)(2) x (2)(2)(2)(2)(2)
2
5
= (2)(2)(2)(2)(2)(2)(2)
= 2
7
2 ÷ 2 = (2)(2)(2)(2)(2)
2
5
(2)(2)
(2 ) =
3
2
2
2
x
2
2
= (2)(2)(2)
= 2
3
2
2
x
= 2
2+2+2
= 2
6
Since the bases are the same, we add the powers: 2 + 5 = 7
So we can write it as:
With bases the same, we subtract the powers: 5 -2 = 3
So we can write it as:
With bases the same, we multiply the powers: 2 x 3 = 6
So we can write it as:
2
7
2
3
2
6
Fractional Exponents
Rule for Fractional Exponents:
Negative Exponents
Rule for Negative Exponents:
The fractional exponent becomes a radical.
The denominator becomes the index.
The numerator becomes the exponent:
A negative exponent becomes a reciprocal fraction.
The negative exponent becomes a positive exponent in the denominator.
27
1
3
√27
3
1
√27
3
= 3
3
3
3
3
-2
3
1
2
9
1
ROOTS & RADICALS
Intro to Roots
√81
4
Radical
Index
Radicand
√81
4
3 3 3 3
Step 1) Since the index is "4" ask yourself, what factor of 81 can be taken out four times;
81 can be divided by 3, four times
Step 2) If you find a factor that can be taken out perfectly, that factor is your answer.
√81= 3
4
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Example 2) What if the radicand cannot be factored perfectly?
52 ÷ 2 = 26
26 ÷ 2 = 13
Circle the factors that appear twice because the index is 2.
The "2's" come outside of the radical and the left over "13" remains inside the radical
√52
2
2 26
2 13
2
√13
2
We say, "two root thirteen," or "two times the square root of thirteen."
Simplifying Radicals
Changing Entire Radicals to Mixed Radicals:
Example 1) Simplify this radical:
√144
3
√144
3
√12 x 12
3
2
2
3
2
2
3
√(2*2*2)(2*3*3)
3
2√(2*3*3)
3
Step 1) Since the index is 3, we are looking for a factor of 144 that can be taken out three times.
Step 2) Circle the factors and put them outside of the radical
Step 3) Re-write the radical with the 2 on the outside and expand the inside to 18
2√18 =
3
√144
3
We say, "two root 18," or "two times the cubed root of eighteen."
Mixed to Entire Radicals
Step 2) Now all numbers are inside the radical, we can simplify.
Step 3) Further simplify
Example 1) Expand this radical:
2√18
3
Step 1) The index (3) indicates the 2 on the outside needs to be be put to the power of 3 in order to go into the radical.
2√18
3
(2*2*2)√18
3
√18*(2*2*2)
3
√18*8
3
√144
3
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