Download Simplifying Radicals

Rational
Exponents &
Radicals
Mrs. Daniel- Algebra 1
Exponents
Definition: Exponent
• The exponent of a number says how many
times to use that number in a multiplication.
It is written as a small number to the right
and above the base number.
The Zero Exponent Rule
• Any number (excluding zero) to the zero
power is always equal to one.
• Examples:
 1000=1
 1470=1
 550 =1
Negative Power Rule
Let’s Practice…
1. 5-2
2.
3 −1
4
3. (-3)-3
The One Exponent Rule
• Any number (excluding zero) to the first power
is always equal to that number.
• Examples:
 a1 = a
 71 = 7
 531 = 53
The Power Rule
(Powers to Powers)
When an exponential expression is raised to a
power, multiply the exponents.
Try these…
1. (w4)2
3. (x3)4
2. (q2)8
Products to Powers
(ab)n = anbn
Distribute the exponent/power to all
variables and/or coefficients.
For example:
(6y) 2 = (62)(y2)= 36y2
(7x3)2 = (7)2(x3)2 = 49x6
Let’s Practice…
1. (5x2)2
4. (6x4)2
2. (3wk3)3
5. (n5)2(4mn-2)3
3. (-2y)4
6. (5x2)2
The Quotient Rule
Let’s Practice…
Power of a Fraction
Let’s Practice…
3.
3 −2
4
Untangling Exponential
Expressions
1. Move expressions with negative outside
exponents to bottom.
2. Distribute all outside exponents.
3. Add/Subtract to combine duplicate
variables.
4. No negatives in final answer!!
Let’s Practice #1
Let’s Practice #2
Let’s Practice #3
Simplifying
Radicals
Radical Vocab
How to Simplify Radicals
1. Make a factor tree of the radicand.
2. Circle all final factor pairs.
3. All circled pairs move outside the radical and
become single value.
4. Multiply all values outside radical.
5. Multiply all final factors that were not circled.
Place product under radical sign.
Let’s Practice…
1. 225
2. 300
Let’s Practice…
3.
1
49
4. 120
How to Simplify Cubed
Radicals
1. Make a factor tree of the radicand.
2. Circle all final factor groups of three.
3. All circled groups of three move outside the
radical and become single value.
4. Multiply all values outside radical.
5. Multiply all final factors that were not circled.
Place product under radical sign.
Let’s Practice…
3
1. 375
3
2. 64
Let’s Practice…
3
3. 81
3
4. 256
Simplifying
Rational
Exponents
Review: Radical Vocab
How to Simplify Radicals
1. Make a factor tree of the radicand.
2. Circle all final factor “nth groups”.
3. All circled “nth group” move outside the
radical and become single value.
4. Multiply all values outside radical.
5. Multiple all final factors that were not circle.
Place product under radical sign.
Let’s Practice…
Simplify:
5
1. 243
4
2. 256
3
3. 135
Code: Fractional Exponents
Let’s Practice #1
Rewrite in exponential form. Simplify if possible.
5
1. 𝑥 2
6
2. 𝑥 3
5
3. 32𝑥 3
3
4. 27𝑑5
4
5. 256𝑎8
Let’s Practice #2
Rewrite as radical expressions, then simplify, if
possible:
1. 12𝑎
2. 6𝑥
2
3
5
2
3. 64𝑎
4
5
Let’s Practice #3
Applications
Applications
Applications
Applications
Rational &
Irrational
Numbers
Rational Numbers
• Any number that can be expressed as the
𝑝
quotient or fraction of two integers.
𝑞
• YES:
– Any integers
– Any decimals that
ends or repeats
– Any fraction
• NO:
– Never ending decimals
Irrational Numbers
• Any number that can not be expressed as a
fraction.
• Usually a never-ending, non-repeating
decimal.
• Examples:
𝜋
2, 5
1.2658945625692….
Let’s Practice…
Rational or Irrational.
1.
2.
2
17
1
3
3. 0
Will it be Rational or
Irrational?
Sums:
Rational + Rational =
Rational + Irrational =
Irrational + Irrational =
Products:
Rational x Rational =
Rational x Irrational =
Irrational x Irrational =