nth Roots and Rational Exponents Solve Radical Equations Objectives: 1.To simplify expressions involving th roots and rational exponents 2.To solve really cool (radical) equations
Dec 24, 2015
nth Roots and Rational ExponentsSolve Radical Equations
Objectives:
1. To simplify expressions involving th roots and rational exponents
2. To solve really cool (radical) equations
Objective 1You will be able to simplify expressions involving th roots and rational exponents
πΈπ§=πππ§
(π§ 2+π 2 )3/2
Electric Field due to a Ring of Charge
Square Roots and Beyond
The number is a square root of if .β’ This is usually written
Radicand
Radical
Square Roots and Beyond
The number is a square root of if .β’ This is usually written
Any positive number has two
real square roots, one positive and
one negative, and
and , since 22 = 4 and (β2)2 = 4
The positive square root is considered the principal square root
Square Roots and Beyond
Likewise, is the cube root of if .β’ This is usually written
Any positive or negative number has one real cube root, and the other two are imaginary
, since
, since
Square Roots and Beyond
Finally, is the th root of if .β’ This is usually written
Index
On a your calculator, cube and th roots can be found in the MATH
Menu
Rational Exponents
Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.
(βπ)2=π
(ππ₯ )2=π
Letππ₯=βπSoπ1/2=βπ
π2π₯=ππ2π₯=π1 2 π₯=1
π₯=12
Rational Exponents
Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.
(ππ₯ )3=ππ3π₯=ππ3π₯=π1 3 π₯=1
π₯=13
( 3βπ)3=π Letππ₯= 3βπSoπ1 /3=3βπ
Rational Exponents
Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.
(π π₯)π=π
πππ₯=ππππ₯=π1 ππ₯=1
π₯=1π
(πβπ )π=π Let ππ₯=πβπSoπ1/π=πβπ
Objective 1You will be able to simplify expressions involving th roots and rational exponents
πΈπ§=πππ§
(π§ 2+π 2 )3/2
Electric Field due to a Ring of Charge
Real nth Roots
In other words, even roots have two solutions, a positive and negative, and the
radicands have to be nonnegative.
Real nth Roots
Furthermore, odd roots only have one solution, with the same sign as the radicand,
which can be positive or negative.
More Rational Exponents
Exponents can be any rational number: a positive or a negative, a proper or an improper fraction.
Let be an th root of , and let be a positive integer.
Solving Equations
Recall the inverse of squaring a number is taking the square root.
Similarly, the inverse of raising a number to the nth power is taking the
nth root.
We can use this relationship to solve certain equations involving th powers.
Really Cool Equations
The equations below are all examples of radical equations.
The radicals involved can be of any index or can even use rational
exponents.
β5 π₯+1=63βπ₯β10=β3
π₯β6=β3π₯β 4 π₯+1=βπ₯+10
Really Cool Equations
The equations below are all examples of radical equations.
To solve these awesome equations, you first have to
isolate the radical expression, and then raise both sides of the equation
to some power to make the radical mathemagically
disappear.
β5 π₯+1=63βπ₯β10=β3
π₯β6=β3π₯β 4 π₯+1=βπ₯+10
Step 3
Step 2
Step 1
Solving Radical Equations
To solve radical equations:
Isolate the
radical Raise each side to some power
Solve new polynomial equation
Square or cube both sidesβ¦
Extraneous Solutions
As the previous exercise demonstrated, it is important to check your solutions because at least one of them may be extraneous. This means that it is an apparent solution that doesnβt actually work in the original equation.
Before squaring:
β1=1
After squaring:
(β1 )2=12
1=1
Step 1
Step 3 Step 2
Step 1
Squaring Ad Nauseum
Really radical equations contain more than one radical expression. To solve these equations:
Separate radicals
Square both sides
Isolate radical
Solve and
check
Rational Exponents
Solving equations involving rational exponents is similar to solving radical equations.
1. Isolate the variable/expression with rational exponents
2. Raise both sides to the reciprocal power
3. Solve and check your answer(s)
(π₯β4 )2/3β9=16
Substitution
Sometimes, using a clever substitution for an expression with a rational exponent can simplify solving an equation. For example, for the equation
we can let . Then the equation becomes(3 π₯+1 )2/3+3 (3 π₯+1 )1/3+2=0
π2+3π+2=0