Unit 2~ the Powers That Be
M8N1.i- Simplify expressions containing exponents
Multiplying exponents:
*When multiplying exponents with the same base, add the
exponents and keep the same base.
Example 1:
8
5

8 9 = 8 14
Keep the base of 8 and subtract -5 – 9 = - 14
(Be sure to follow the integer rules.)
Example 2:
76

74 = 72
Keep the base of 7 and subtract 6 – 4 = 2 (Be
sure to follow the integer rules.)
Power to a power:
6
*When an exponent is taken to an exponent. (4 3 ) Multiply
the two exponents and keep the same base. 3  6 = 18 keep the
same base of 4. So the exponential form would be 4 18 .
Dividing exponents:
*When dividing exponents with the same base, subtract the
exponents and keep the same base.
Example 1:
87
 83
4
8
a5
Example 2: 9  a 4
a
Keep the same base and subtract 7- 4= 3
Keep the same base and subtract 5 – 9 = -4 (remember to
follow integer rules)
Negative exponents:
*When changing negative exponents to positive exponents.
Take the reciprocal (turn upside down) and change the exponent
positive.
1
Example 1: 3 5 = 5 Make it a fraction and change the exponent positive.
3
Example 2:
2
8x 6

8
3x 2
12 x
Simplify
8 4
2
 
12 4
3
then subtract the exponents with
the base of x, 6 – 8 = -2. Since the exponent is negative the variable will be in the
denominator.
Exponent of zero
*Any base with an exponent of zero equals 1, except 0 0 = 0.
Example1: 5 0 = 1
Example2: y 0 = 1
Example3: 743 0 =1
Scientific Notation
M8N1.j~ Express and use numbers in scientific notation.
 Scientific Notation is used to write very large or very small
numbers.
 Scientific Notation deals with decimal placement. (It has
nothing to do with the amount of zeros)
 The exponent tells how many times to move the decimal.
*If the exponent is positive the number in standard
form will be greater than 1.
*If the exponent is negative the number in
standard form will be less than 1 (meaning
it will be a decimal number).
Converting from standard to scientific notation
 Every number has a decimal. If the decimal is not
shown it is behind the ones place.
To change a number from standard form to scientific notation:
- Step 1: Move the decimal so that there is one
non-zero digit in front of the decimal.
- Step 2: Count how many decimal placements you
had to move the decimal. This number will be
your exponent; the base will always be ten.
Example 1:
352,000,000 ---------- 3.52 X 10 8
Since the number is greater
than 1, the exponent is positive.
Example 2:
0.00003 ---------- 3 x 10 5
Since the number is less than 1,
the exponent is negative.
Example 3:
0.00078 ------- 7.8 X 10 4
exponent is negative.
Since the number is less than 1, the
Converting from scientific notation to standard
-
Step 1: If the exponent is positive move the decimal to
the right the number of exponent times
If the exponent is negative move the decimal to
the left the number of exponent times
Example 1: 7.94 X 10 9 = 7,940,000,000
Example 2: 9.105 X 10 7 = 0.0000009105
Example 3: - 4.8 X 10 8 = -480,000,000
Multiplying scientific notation
-
Step 1: Multiply the leading two number
Step 2: Add the exponents and keep the base of 10
Step 3: Be sure the number is in scientific notation.
(Meaning it is one non-zero digit in front of the decimal)
If not be sure to adjust the exponent based on the
decimal movement.
Dividing Scientific notation
-
Step 1: Dividing the leading two number
Step 2: Subtract the exponents and keep the base of 10
Step 3: Be sure the number is in scientific notation.
(Meaning it is one non-zero digit in front of the decimal)
If not be sure to adjust the exponent based on the
decimal movement.
8.4 X 10 7
 4 X 10 3 Divide 2.1 8.4 move the decimal
4
2.1X 10
because you can mot divide by a decimal number. So the
problem becomes 21 84 = 4. Then subtract the
Example1:
denominator’s exponent from the numerator’s exponent
7- 4 =3.
7.5 X 10 8
 6.25 X 10 14 Divide 1.2 7.5  0.625 then
5
1.2 X 10
subtract the exponents -8 – 5 remember to use your integer
rules -8 – 5 = -13. Write your answers 0.625 X 10 13 , this is
NOT in scientific notation. Move the decimal back one
place 6.25 then subtract one from the exponent -13 -1= -14
So the final answer is 6.25X10 14
Example 2:
M8N1.h~ Distinguish between rational and irrational numbers
Rational or Irrational
Rational Numbers
 Any Number that can
be written as a
fraction

(note: the sign of the
number does not matter it
can be negative or positive)
Examples:
5
7
2
5 since this mixed fraction can be
3
changed to an improper fraction it is rational.
3 since a whole number can be written as a
fraction it is a rational number.
 Any Decimal that
terminates (stops)
 Any Decimal that
repeats the same digit
or the same series of
numbers
 Any perfect square
radical
Irrational Numbers
 Any number that never
stops and repeats.
 Any non-perfect
square radical
Radicals:
7.34
2.5678
5. 4
3.181818….
36
100
Examples:
 6.45972530163…….
18
32
M8n1.a ~Find square roots of perfect squares.
Perfect Square number means the number has two same factors.
Example: 3  3 = 9 Therefore 9 is a perfect square which equals 3.
3 is the square root of 9 .
This symbol is called a radical. The number under the radical is
called a radicand.
Every Perfect Square Radical has two roots. The Negative and the
Positive root. The Positive root is called the Principal root.
49 = +7 principal root and -7
M8N1.f-Estimate square roots of positive numbers
Estimating Radicals
Best Whole number estimate of a square.
Step 1 – Find the closest perfect square less than the radical.
Step 2 -- Find the closest perfect square greater than the radical.
Step 3 – Subtract the difference from each radical, to find the radical closest.
Step 4 – Take the square root of the closet radical and that is the closest
whole number
EXAMPLE:
32
Since the 25 is seven away from the 32 and
is the four away from the 32 , the closest whole
number is 6.
36
25
36
M8N1.i~Simplify, add, subtract, multiply, and
divide expressions containing square roots.
Simplifying square roots
Step 1 – Make a prime factorization tree for the radicand. Circle all the
prime numbers.
Step 2 – Identify twin prime numbers (any two prime numbers) cross
them both out then write one as a whole number.
Step 3 – Any prime numbers that are left multiply and leave under the
radical.
**For every set of twins take one out and leave the rest in, under
the radical.
Examples:
32
3 48
ADDING and SUBTRACTING RADICALS
*In order to add or subtract radicals that radicals must be the same, just like
when adding or subtracting fractions.
Step 1: Simplify each of the radicals.
Step 2: If the radicals are the same add or subtract the whole numbers in
front of the radical. (Remember that the coefficient is 1 if there is not a whole
number in front.)
Step 3: Keep the same radical
Examples:
Multiplying radicals
Step 1–Simplify the radicals, using prime factorization trees
Step 2 – Identify the twin prime numbers in both of the
numbers. For each twins cross them out then write one as a
whole number.
Step 3 – Any prime numbers left multiply them then write
them under a radical.
Step 4 – Be sure to bring down any whole numbers, then
multiply the whole numbers.
Examples:
Dividing Radicals:
Step 1: Simply the numerator and denominator.
Step 2: Determine if the denominator can divide into the numerator.
*If yes divide and simplify.
*If no then you must rationalize the denominator. This means
multiply the numerator and the denominator by the radical in the
denominator.
Step 3: Simplify the numerators’ radical
Step 4: Simplify any fractions.
EXAMPLES:
M8G2.a~ Apply properties of right triangles, including the
Pythagorean Theorem
M8G2.b~Recongnize and interpret the Pythagorean
Theorem as a statement about areas of squares on the side
of a right triangle.
Pythagorean Theorem:
a2 + b2 = c2
*Hypotenuse – side of a right triangle that is opposite the
right angle
*Legs the two shorter sides of a right triangle.
To find the hypotenuse when given both leg lengths:
Step 1: begin with the formula a 2 + b 2 = c 2 . Replace the a
and b with the correct measurements that were given.
Step 2: Square both of the measurements then find the sum
(add).
Step3: Take the square root of c 2 . Which will give you c.
Then take the square root of the sum. If the radical is not a
perfect square then simplify the radical or estimate.
Example:
To find a leg measurement when given one leg and the
hypotenuse:
Step 1: begin with the formula a 2 + b 2 = c 2 . Replace the
measurement of the leg with a or b. Then replace c with the
measurement of the hypotenuse.
Step 2: Square both of the measurements.
Step 3: Subtract the leg measurement from the hypotenuse
measurement.
Step 4: Take the square root on both sides of the equal sign.
Simplify the radical.
EXAMPLES: