Download Simplifying Non-Perfect Square Radicals

Radical Expressions Containing Variables
- To simplify a radical expression with variables:
--If the exponent is even, divide the exponent by
two, and that will be your new exponent
Ex. √𝑥 12 = √(𝑥 6 )2 = 𝑥 6
Ex. √𝑑 4 = √(𝑑 2 )2
= 𝑑2
Ex. √𝑥 24 =
Ex. √𝑎16 =
--If the exponent is odd, separate into two
terms. The first term will be the exponent
inside the radical sign. The second term will be
the exponent one less divided by 2
Ex. √𝑥 5 = √𝑥 4 (𝑥 1 ) = 𝑥 2 √𝑥
Ex. √𝑎11 = √𝑎10 (𝑎1 ) = 𝑎5 √𝑎
Ex. √𝑑15 =
Ex. √𝑥 23 =
- If the square root of a variable raised to an even power
has a variable raised to an odd power for an answer, the
answer must have absolute value signs. This ensures that
the answer will be positive.
Ex. √𝑥 6 = √(𝑥 3 )2 = |𝑥 3 | = |𝑥 |3
Ex. √(−2)10 = √(−25 )2 = |(−2)5 | = 25
Ex. √𝑥 14 =
Ex. √𝑓 22 =
Simplifying Non-Perfect Square Radicals
- To simplify a radicand, first separate the number into
two factors where one factor is a perfect square.
- Then, simplify the perfect square and keep the second
factor as a radical. If there is a radical, the number is
considered irrational.
- If after factoring you notice that you could have
another perfect square, factor again until there are no
perfect squares left!
- If there is a number outside of a radical, multiply the
number to any simplified perfect squares.
Ex. √8 = √4√2 = 2√2
Ex. √125 = √25√5 = 5√5
Ex. √72 =
Ex. √18 =
Ex. √360 =
Ex. √24 =
Ex. √108 =
Ex. 2√24 =
Simplifying Roots of Variables
- We will now combine everything we did on the first
two lessons.
- With variables, divide the exponent by 2. The number
of times that 2 goes into the exponent becomes the
power on the outside of the radical and the remainder is
the power of the radicand.
√𝑥 7 = 𝑥 3 √𝑥
Note: Absolute value signs are not needed because the
radicand had an odd power to start.
- With numbers, simplify by simplifying the perfect
square out of the radicand and keeping the remaining
factor inside the radical sign.
- Combine both answers.
Ex. √50𝑥 4 𝑦12 𝑧 3 = √25√2√𝑥 4 √𝑦12 √𝑧 2 √𝑧
=
5 √2 𝑥 2 𝑦 6
= 5𝑥 2 𝑦 6 𝑧√2𝑧
Ex. √8𝑥 5 𝑦 6 𝑧 4 =
𝑧 √𝑧
Ex. √16𝑥 7 𝑦 4 𝑧 =
Ex. √𝑚11 𝑛22 =
Ex. √40𝑐14 𝑑 7 =
Ex. √100𝑥 3 𝑦10 𝑧 4 =
Solving Equations with Perfect Squares and
Cube Roots
- The product of two “equal” factors is the SQUARE of
the number
(5)(5) = 25
- The product of three “equal” factors is the CUBE of the
number
(5)(5)(5) = 125
- When solving equations, you need to find the proper
root based on what the question is asking.
Note: If the root you are looking for is square, you must
show a ± sign to show the root could be positive or
negative!
Note: If the CUBE is negative, then your root is also
negative!
Ex. 5𝑥 2 = 605
5𝑥 2
5
=
605
5
𝑥 2 = 121
√𝑥 2 = √121
𝑥 = ±11
Ex.
𝑥3
9
𝑥3
9
= −3
(9) = −3(9)
𝑥 3 = −27
3
√𝑥 3 = √−27
𝑥 = −3
Ex. 4𝑥 3 = 500
Ex.
𝑥2
8
= 18