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Formulas for Exponent and Radicals
Algebraic Rules for Manipulating Exponential and Radicals Expressions.
In the following, n, m, k, j are arbitrary -
. they can be integers or rationals or real numbers.
bn^ · bm
bk^
= bn+m−k^ Add exponents in the numerator and
Subtract exponent in denominator.
( an^ · bm ck
)j
=
an·j^ · bm·j ck·j^
The exponent outside the parentheses
Multiplies the exponents inside.
( an bm
bm an^
Negative exponent ”flips” a fraction.
b^0 = 1 b = b^1 Don’t forget these
Convert Radicals to Exponent notation
√ a = a^1 /^2 √ ma = a 1 /m √ man (^) = an/m
Radicals - Reducing
a^2 · b = a
b Remove squares from inside √ m am^ · b = a
√m b
Exponent and Radicals - Solving Equations
Solve a power by a root xn/m^ = y ⇔ x = ym/n^ Solve a root by a power
Example
a) Simplify
Method
b) Simplify
Method
22 · 32 ·^2
53 ·^2
Illustration: where is the negative?
c) Simplify
( the ’negative’ is inside the parentheses)
Method
d) Simplify −
( the ’negative’ is outside the parentheses)
Method −
e) Simplify
( the ’negative’ is in the exponent)
Method
(2/5)^3
= (or =
More Examples with negatives
i) Simplify
6 st−^4
2 s−^2 t^2
(give answer with only positive exponents )
Negative exponents flip location: A negative exponent in the numerator moves to the denominator. And a negative exponent in the denominator moves to the numerator.
Method 6 st
− 4
2 s−^2 t^2
6 ss^2 2 t^4 t^2
3s^3 t^6
j) Simplify
y 3 z^3
(give answer with only positive exponents )
A Negative exponent ’flips’ the fraction.
Method
y
3 z^3
3 z^3
y
9z^6
y^2
More Examples
k) Simplify
(2x^3 )^2 (3x^4 )
(x^3 )^4
Method (2x
(^3) ) (^2) (3x (^4) )
(x^3 )^4
22 x^3 ·^2 · 3 x^4 x^3 ·^4
(4 · 3)x^6 x^4 x^12
= 12x6+4−^12 =
x^2
More Examples
e) Simplify
Method
f) Simplify
25 b −
b^3
Method
25 b −
b^3 =
25 · b +
b^2 · b
= 5
b − b
b = ( 5 − b)
b
Exponents and Radicals
Evaluate the Expression (negative exponents) - without using a calculator
a) − 2 −^2 b) (−2)−^2
c)
2 −^3
d)
3 −^1
e) 6−^1 + 5−^1 f) − 1 −^1 · (−2)−^2
Simplify each Expression (integer exponents)
a) (− 3 x^2 y^3 )(2x^9 y^8 ) b) (− 6 a^7 b^4 )(3a^3 b^5 )
c) x^2 x^4 + x^3 x^3 d) (− 2 b^2 )(3b^3 ) + (5b^3 )(− 3 b^2 )
e) (−m^2 )(−m) − m(−m) + m(3m^2 ) f) (z^2 )(−z) − (−z) − z(−z^2 ) + z(2z)
Answers a) -1/4; b) 1/4; c) 8; d) 1/24; e) 11/30; f) -1/4; Answers a) − 6 x^11 y^11 ; b) − 18 a^10 b^9 c) 2x^6 ; d) − 21 b^5 ; e) 4m^3 + m^2 ; f) 2z^2 + z
Radicals Simplify without a calculator - then check using a calculator.
a) − 91 /^2 b) (−27)^4 /^3
c) 8−^4 /^3 d)
e)
163 f)
Simplify each expression (ignore absolute value at this time.)
a) (a^15 )^1 /^5 b) (x^6 )^1 /^6
c) (x^3 y^6 )^1 /^3 d) (16x^4 y^8 )^1 /^4
Simplify each expression - write answers without negative exponents.
a)
6 a^1 /^2 2 a^1 /^3
b)
− 4 y 2 y^2 /^3
c) (a^2 b^1 /^2 )(a^1 /^3 b−^1 /^2 ) d)
x^1 /^2 y y^1 /^2
Answers a) −3; b) 81; c) 1/16; d) 8/27; e) 8; f) 32 Answers a) a^3 ; b) |x|; c) xy^2 ; d) 2|x|y^2 ; Answers a) 3a^1 /^6 ; b) − 2 y^1 /^3 ; c) a^7 /^3 ; d) x^3 /^2 y^3 /^2 ;
Change Notation radical-exponent - use only positive exponents
a) a(b^4 + 1)−^1 /^2 b) − 23 /^4
c)
x^3 d) 3
x^3 + y^3
Simplify each radical expression (assume everything is positive.)
a)
16 x^2 b)
xy 100
c)
3
− 8 a^3 b^15
d) 4
16 t^4 y^8
Rationalize the denominator
a)
b)
c)
√ (^3) x d)
x y^2 /^5
e)
f)
x −
y
Answers a) √ba (^4) +1 ; b) − 4
√ 23 ; c) x^3 /^5 ; d) (x^3 + y^3 )^1 /^3 ;
Answers a) 4x; b)
√xy 10 ; c)^
− 2 a b^5 ; d)^
2 t y^2 ; Answers a)
√ 10 10 ; b)
√ 15 6 ; c)^
2 3
√ x^2 x ; d)^
xy^3 /^5 y ; e)^
10+2√ 6 19 ; )
√x+√y x−y ;
Solve the rational exponent problems for x.
a) 4x^1 /^3 = 20 b) 3x^1 /^4 = 15
c) 3x^3 /^4 = 24 d) 4x^1 /^3 + 20 = 0
Solve the rational exponent problems for x.
a) x^4 /^3 − 16 = 0 b) 3x^5 /^3 + 96 = 0
c) (x − 3)^3 /^2 = 27 d) (x − 7)^4 = 16
Answers a) 125; b) 625; c) 16; d) − 125 Answers a) ±8; b) −8; c) x = 12; d) 9, 5
More Practice
1a) Simplify the expression:
3 t^4 x^4
8 t^7 x^3
1b) Solve for x: 5x^5 /^3 + 60 = 38940
2a. Simplify the exponential expression:
x^11 y^19
x^10 y^5
x−^137 y^4
2b. Evaluate the expression at x = 22, y = 11:
x^11 y^19
x^10 y^5
x−^137 y^4
- Simplify the variable exponential expression:
xt−^3
x^2 t−^2
- Rationalize the denominator:
- Simplify the radical expression: 14
6a. Simplify the radical expression:
6b. Simplify the radical expression:
80 x^10 +
180 x^10
Answers
1a) 24t^11 x^7 1b) 6^3 = 216 2 a) x
2 y^4 ;^ b)^
4 121 3)^ x
6 t− (^10) 4) 32+
√ 3 22 =^
1 22
( 32 + 24
√ 3
)
- 112
√ 5 6a) 10
√ 5 6b) 10x^5
√ 5
