Download CCSS N-RN.1 Rational Exponents

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CCSS N-RN.1 Rational Exponents
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June 11, 2013
CCSS N-RN.1 Algebra – Rational Exponents
This lesson on simplifying rational exponents is intended to align with
the Common Core (CCSS) High School Algebra standards. Like all
common core standards that build on material from previous courses,
this builds on integer exponents and radicals from 8th grade Algebra.
Intended Course(s): High School Algebra 1, Algebra 2
Common Core Code: N-RN.1 Rational Exponents
http://www.corestandards.org/Math/Content/HSN/RN
CCSS.Math.Content. HSN- RN.A.1
“Explain how the definition of the meaning of rational exponents follows from extending
the properties of integer exponents to those values, allowing for a notation for radicals in
terms of rational exponents. For example, we define 51/3 to be the cube root of 5
because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.”
Duration: 2 Days (It’s flexible, depending on what you keep and what
you edit out.)
Objectives: The student will simplify integer exponents (review).
The student will convert rational exponents into radicals and simplify.
The student will convert radicals into rational exponents and simplify.
Possible questions to ask students to build a deeper understanding:
Dig Deeper Questions:
Why is (-3)2 = 9 but -32 = -9?
Why is there a stipulation on the Zero Exponent that a cannot
equal zero?
Simplify: 2-1 + 3-2
1 3/2
Ask students to evaluate (16)
.
How would you write the following using rational exponents?
5
√(3𝑥 − 4)2
Place the following real numbers in order from least to
greatest.
93/2,
1
5−2
, 170 , 36-3/2
Lesson Introduction:
People, teens especially, are great at using shorthand abbreviations for
texting. I believe it is called “social shorthand.” For example, BFF
means ? (best friend forever), ROTFL means ? (rolling on the floor
laughing), and SETE means ? (smiling ear-to-ear).
Well, mathematicians also use their own kind of shorthand. The
purpose is the same as yours, who wants to write all of that out. Here’s
an example:
35 which means 3∙3∙3∙3∙3 or 243
The 3 is the base and the 5 is the exponent or power. Remember in
math you have to have the eyes of an eagle and take notice of the
tiniest detail.
For example:
(3x)5 and 3x5
Are the bases the same? (No, 3x and x) What is the 3 called in 3x5?
(coefficient)
An exponent is just a way of showing how many times a number is
multiplied by itself.
From previous math courses:
You were first introduced to natural number exponents. You learned
that if the exponent was 2 it was squared and if it was 3 it was cubed.
Evaluate a. 42 b. (-2)3 c. -14 d. (1/2)3
a. 16 b. -8 c. -1 d. 1/8
Dig Deeper Question:
Why is (-3)2 = 9 but -32 = -9?
Answer: (-3)2 = (-3)(-3) the base is (-3) but -32 = - 3 ∙ 3 = -9 the base is
3.
Negative Exponents:
Next, you probably learned about negative exponents.
Remember negative exponents do not
make numbers positive or negative, they
are like elevators that move things.
In 2002, after their hit “Who Let the Dogs out?” the Baha Men came
out with the song “Can you move it like this?” This reminds me of
negative exponents, because negative exponents “move things.”
For example: If you imagine
4-1
written as a fraction
4−1
1
, the negative
exponent moves it to the denominator. Once you move it, you take
away the negative sign of the exponent.
-1
4
1
But if it’s
4 −1
1
=
or
41
1
4
it moves it above, into the numerator. It’s like an
elevator.
1
= 41 or 4
4 −1
Evaluate:
a. 2
-1
-1
b. 3
c.
1
7−1
d.
1
8−1
a. ½ b. 1/3 c. 7 d. 8
Evaluate:
a. 3
-2
-3
b. 4
a. 1/9 b. 1/64 c. 1/25 d. 32
c. 5
-2
d.
1
2−5
Dig Deeper Question:
Simplify: 2-1 + 3-2
𝟏
Answer:
𝟐
+
𝟏
𝟗
=
𝟗
𝟏𝟖
+
𝟐
𝟏𝟖
=
𝟏𝟏
𝟏𝟖
Zero Exponent
You may have learned about zero as an exponent.
If a does not equal 0, then a0 = 1.
Evaluate:
a. 50
b. -50
c. (x + 5)0
d. 5x0
a. 1 b. -1 c. 1 d. 5
Do you recall the rules for exponents?
Quotient Rule for Exponents:
If a is a nonzero real number and n and m are both integers then,
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
5
5
Why is 50 = 1? We know that = 1.
Try it in your calculator 5 divided by 5 = 1.
If we use the quotient rule for exponents on
5
5
which is
51
51
it
would be 51-1 which is 50 which must be 1.
Dig Deeper Question:
Why is there a stipulation on the Zero
Exponent that a cannot equal zero?
Answer: Plug zero in like the
𝟎𝟏
𝟎𝟏
𝟓
𝟓
and see what happens.
this is undefined because you can’t divide by zero.
Note: You might want to tell students that one of the tools we use to find the answer to these
types of questions, is to just plug it in and see where it takes us.
Evaluate:
0
a. -3x + y
a. - 3 + 1 = -2
0
b. -7a
b. -7(1) = -7
0
c.
−5𝑥𝑦−4𝑧 0
-(
)
2𝑦𝑧
c. -1
Radicals
Up until this point you have used exponents that were integers, but
where do radicals fit in? What is a radical?
Just as subtraction is the opposite of addition, a radical or root of a
number is the opposite of an exponent.
http://www.purplemath.com/modules/radicals.htm
"Roots" (or "radicals") are the "opposite" operation of applying
exponents; you can "undo" a power with a radical, and a radical
can "undo" a power. For instance, if you square 2, you get 4, and
if you "take the square root of 4", you get 2; if you square 3, you
get 9, and if you "take the square root of 9", you get 3.” Copyright
© Elizabeth
There are three important parts. The symbol is called the radical
(actually just the check mark part), the 8 is the radicand, and the 3 is
the index. This is the cube root of 8 and means what number times
itself and times itself again equals 8? The answer is 2 because 23 = 8.
3
√8
Rational Exponents (new material):
Definition of a1/n
𝑛
If n is a positive integer greater than 1 and √𝑎
is a real number, then
𝑛
𝑎1/𝑛 = √𝑎
Here are a few examples.
2
41/2 = √4 which is 2.
2
91/2 = √9 which is 3.
1/2
16
2
= √16 which is 4.
Anything to the one-half power is the same as
taking the square root of the number.
Remember we don’t need to write the index of 2. It is understood.
How about cube roots?
3
81/3 = √8 which is 2 because 23 = 8.
3
271/3 = √27 which is 3 because 33 = 27.
3
641/3 = √64 which is 4 because 43 = 64.
Anything taken to the one-third power is the
same as taking the cube root of the number.
What about other roots?
1/4
81
4
= √81 = 3 because 34 = 81.
5
321/5 = √32 = 2 because 25 = 32.
The same applies to the fourth root, fifth root,
etc.
Rewrite the following radicals as rational
exponents.
4
a. √5
a. 51/4
6
b. √235
b. 2351/6
c. 10071/2
c. √1007
Notice how roots undo exponents?
Definition of am/n
If m and n are positive integers greater than 1,
and
𝒎
𝒏
is reduced, then
a
m/n
=
𝒏
√𝒂𝒎
𝒏
𝒐𝒓 ( √𝒂)
𝒎
𝒏
as long as √𝒂 is a real number.
For example:
2/3
8
3
2
= ( √8) simplify inside ( ) first. (2)2 = 4
You may want to show students how to check
this on their calculators 8^(2÷ 3) enter.
2/3
2
3
(-27) = ( √−27) = (-3)2 = 9. You may want
to show students that often the other way is
3
3
more difficult. ( √(−27)2 ) = √729 = 9
In a calculator (-27)^(2÷ 3) enter.
3/5
-32
3
5
= -( √32) = -(2)3 = -8
In a calculator -32^(3÷ 5) enter.
5
4
How would you write ( √625) with a rational
exponent? Simplify the expression.
Answer: 6255/4 and 3125
Dig Deeper Question:
Ask students to evaluate
𝟑
𝟐
𝟏
𝟏 𝟑
Answer ( √ ) = ( ) =
𝟏𝟔
𝟒
𝟏
𝟔𝟒
In a calculator (1÷16)^(3÷ 𝟐) enter.
𝟏 𝟑/𝟐
( ) .
𝟏𝟔
Definition of a-m/n
a
-m/n
=
𝟏
𝒂𝒎/𝒏
as long as am/n is a nonzero real number.
Evaluate:
a. 100-3/2
a.
𝟏
𝟏𝟎𝟎𝟎
𝐛.
𝟏
𝟗
b. (-27)-2/3
c.
c. 81-3/4
𝟏
𝟐𝟕
Dig Deeper Question:
How would you write the following using
rational exponents?
𝟓
√(𝟑𝒙 − 𝟒)𝟐
Answer: (3x-4)2/5
Dig Deeper Question:
Place the following real numbers in order from
least to greatest.
3/2
9 ,
1
5−2
, 170 , 36-3/2
Answer:
36-3/2, 170,
𝟏
𝟓−𝟐
, 93/2 (1/216, 1, 25, 27)
Rewrite the following radicals as rational
exponents.
a.
1
3
√64
a. 64-1/3
b.
b. 253/4
4
√253
c. 49/5
5
c. ( √4)
9
Works Cited:
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Digger clipart - Microsoft
Speed Drill (Easy)
Rewrite the following rational exponents as radicals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
41/2
91/2
161/2
251/2
1441/2
81/3
271/3
641/3
161/4
31251/5
Rewrite the following radicals as rational exponents.
11.
12.
13.
14.
15.
√81
3
√343
5
√−32
8
√256
(√4)
3
Speed Drill Answer Key
Rewrite the following rational exponents as radicals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
41/2
91/2
161/2
251/2
1441/2
81/3
271/3
641/3
161/4
31251/5
√𝟒
√𝟗
√𝟏𝟔
√𝟐𝟓
√𝟏𝟒𝟒
𝟑
√𝟖
𝟑
√𝟐𝟕
𝟑
√𝟔𝟒
𝟒
√𝟏𝟔
𝟓
√𝟑𝟏𝟐𝟓
Rewrite the following radicals as rational exponents.
11.
12.
13.
14.
15.
√81
3
√343
5
√−32
8
√256
(√4)
3
811/2
3431/3
(-32)1/5
2561/8
43/2
Name:________________________ Date:____________
Rewrite the following in radical notation and simplify if possible.
1. 641/2
2. 1251/3
3. (7xy)1/4
4. (-8)-4/3
5. x-1/6
Rewrite the following as a rational exponent.
6. √19
7.
8.
3
√112
1
6
√5
9. (√3)
3
10. - √4
7
Name: Answer Key
Rewrite the following in radical notation and simplify if possible.
1. 641/2 = √𝟔𝟒 = 8
𝟑
2. 1251/3 = √𝟏𝟐𝟓 = 5
3. (7xy)1/4 = 𝟒√𝟕𝒙𝒚
4. (-8)-4/3 =
𝟏
(−𝟖)
𝟒/𝟑 = 𝟑
𝟏
=
𝟒
√(−𝟖)
𝟏
𝟏𝟔
𝟏
5. x-1/6 = 𝟔
√𝒙
Rewrite the following as a rational exponent.
6. √19 = 191/2
7.
8.
3
√112 = 112/3
1
6
√5
= 5-1/6
7
9. (√3) = 37/2
3
10. - √4 = -41/3