Download Roots and Radical Expressions

Roots and Radical
Expressions
Basic Introduction and Rules
The basic definitions of radical expressions:
index
power
𝑎𝑛
=𝑏 ↔
Radical sign
𝑛
radicand
𝑏=𝑎
What are the real-number roots of each radical expression?
3
a) 343
Because the prime factors of 343 are 7*7*7, 7 is the third (cube)
root of 343.
On the graphing calculator, press MATH, 4, 343, Enter to get 7.
b)
4
1
625
What is the prime factorization of 1?
We are looking for groups of 4 (the index), so we get 1*1*1*1.
What is the prime factorization of 625?
When we take the factor tree all the way to all prime numbers,
we get 5*5*5*5.
b)
4
1∗1∗1∗1
5∗5∗5∗5
1
5
=
Since we have one GROUP of four 1’s in the numerator, and one
1
1
GROUP of four 5’s in the denominator, the fourth root of
=
625
5
In this case, we must remember that since we took an even
root, there will be both a positive and a negative answer.
4
1
625
=
1
±
5
Another way to express the fact that there could be two answers
is to use the absolute value sign.
1
5
is the same thing as
1
± .
5
What are the real-number roots of each radical expression?
b)
4
1
625
We MUST use absolute value signs EVERY time that we take an
EVEN root and get an ODD powered answer.
1
th
In this case, we took the 4 root (even) and got to the first
5
power, (odd).
We can leave out the absolute value signs if we are told to
assume that all variables are positive.
What are the real-number roots of each radical expression?
3
c) −0.064
On the graphing calculator, press MATH, 4, -0.064, Enter to get
-0.4.
d) −25
Because 52 = 25 and (−5)2 = 25, neither 5 nor −5 are real
number square roots of −25.
There are no REAL-number square roots of -25.
On the graphing calculator, press 2nd, 𝑥 2 , -25, Enter to get
ERR:NONREAL ANS. Again, there is no real square root of -25.
Find the real-number roots of each radical expression.
1)
4)
3
169
𝟏𝟑
1
−
8
𝟏
−
𝟐
5)
𝑵𝑶𝑵𝑬
8)
−
7)
4
25
10) −0.0001 𝑵𝑶𝑵𝑬
4
2)
11)
3
729
𝟗
3)
𝟐
𝟏𝟏
6)
4
0.1296 𝟎. 𝟔
9)
5
1
243
4
121
𝟏
𝟑
12)
4
0.0016
3
125
216
3
−0.343 −𝟎. 𝟕
3
8
125
𝟎. 𝟐
𝟓
𝟔
𝟐
𝟓
𝑛
You can not assume that 𝑎𝑛 = 𝑎.
For example, (−6)2 = 36 = 6, and −6.
This leads to the following property for any real number 𝑎.
𝑛
If 𝑛 is odd, 𝑎𝑛 = 𝑎
𝑛
If 𝑛 is even, 𝑎𝑛 = 𝑎 .
Again, we use absolute value signs if we took an EVEN power and got an
ODD power answer.
What is the simplified form of each radical expression?
a)
3
1000𝑥 3 𝑦 9
b)
4
256𝑔8
ℎ4 𝑘 16
What is the simplified form of each radical expression?
a)
3
1000𝑥 3 𝑦 9
𝟑
𝟏𝟎 ∗ 𝟏𝟎 ∗ 𝟏𝟎 ∗ 𝒙 ∗ 𝒙 ∗ 𝒙 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚
𝟑
𝟏𝟎 ∗ 𝟏𝟎 ∗ 𝟏𝟎 ∗ 𝒙 ∗ 𝒙 ∗ 𝒙 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚
We have 1 group of 3 10’s, 1 group of 3 x’s, and 3 groups of 3 y’s.
These all come out from inside the radical.
𝟏𝟎 ∗ 𝒙 ∗ 𝒚 ∗ 𝒚 ∗ 𝒚
𝟏𝟎𝒙𝒚𝟑
What is the simplified form of each radical expression?
4
b)
256𝑔8
ℎ4 𝑘 16
𝟒
𝟒
𝒉∗𝒉∗𝒉∗𝒉∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌
𝟒
𝟒
𝟒∗𝟒∗𝟒∗𝟒∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈
𝟒∗𝟒∗𝟒∗𝟒∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈∗𝒈
𝒉∗𝒉∗𝒉∗𝒉∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌∗𝒌
𝟒𝒈𝟐
𝒉 𝒌𝟒
To convert from radical form to rational exponents, place the power of the
radicand in the numerator, over the index.
𝑛
𝑚
𝑛
𝑎𝑚 = 𝑎 .
4
2
For example, 𝑥 4 = 𝑥 = 𝑥 2 .
Rewrite the following radicals into rational exponents.
4
3
4
2
1) 𝑥 𝑦𝑧
2) 𝑥 4 𝑦 5 𝑧 7
𝟒 𝟏 𝟐
𝒙𝟑 𝒚𝟑 𝒛𝟑
4)
𝟒 𝟓 𝟕
𝒙𝟒 𝒚𝟒 𝒛𝟒
𝑎5 𝑏𝑐 2
𝟓 𝟏 𝟐
𝒂𝟐 𝒃𝟐 𝒄𝟐
=
𝟓 𝟏
𝒂𝟐 𝒃𝟐 𝒄
=
𝟓 𝟕
𝒙𝒚𝟒 𝒛𝟒
3)
7
𝑥 14 𝑦𝑧 2
𝟏𝟒 𝟏 𝟐
𝒙 𝟕 𝒚𝟕 𝒛𝟕
=
𝟏 𝟐
𝟐
𝒙 𝒚𝟕 𝒛𝟕
To convert from rational exponent form to radical form, reverse the process.
𝑚
𝑛
𝑎 =
𝑛
𝑎𝑚 .
3
2
For example, 𝑥 = 𝑥 3 .
Rewrite the following rational exponents into radicals.
3
7
1) 𝑥 𝑦
𝟕
5
7
𝒙𝟑 𝒚𝟓
3
4
1 7
4 4
2) 𝑥 𝑦 𝑧
𝟒
𝒙𝟑 𝒚𝒛𝟕
4
5
3 3
10 5
3) 𝑎 𝑏 𝑐
𝟏𝟎
𝒂𝟖 𝒃𝟑 𝒄𝟔
Classwork:
Complete the worksheet on simplifying radicals.
Homework:
Complete the worksheet on converting radicals to rationals and vice-versa.