Online College Courses for Credit

+
3 Tutorials that teach Applying the Properties of Radicals
Take your pick:
Applying the Properties of Radicals

Applying the Properties of Radicals

Author: Sophia Tutorial
Description:

Simplify a radical expression using the properties of radicals.

(more)
See More
Tutorial
what's covered
  1. Properties of Radicals
  2. Cautions when Applying the Properties
  3. Applying the Properties of Radicals

1. Properties of Radicals

There are several properties of radicals we can apply to simplify expressions involving radicals. The following properties are generally true whenever n is greater than 1, and a and b are both positive real numbers:

n-th root of a to the power of n end root equals a and left parenthesis n-th root of a right parenthesis to the power of n equals a

Product Property: n-th root of a b end root equals n-th root of a • n-th root of b

Quotient Property: n-th root of a over b end root equals fraction numerator n-th root of a over denominator n-th root of b end fraction

Fractional Exponents: n-th root of a to the power of m end root equals a to the power of m over n end exponent


2. Cautions when Apply the Properties

Avoid these common errors when applying properties of radicals:

hint
The properties of radicals only apply to factors; they do not apply to terms. For example, we can use the product property of radicals to break square root of 15 into two radicals square root of 5 times square root of 3 because 5 times 3 equals 15. However, we cannot break square root of 8 into square root of 5 plus square root of 3.
hint
We can only bring an exponent outside of a radical if it applies to everything underneath the radical. For example, we can rewrite cube root of x squared end root as left parenthesis cube root of x right parenthesis squared because the exponent of 2 applied to everything underneath the radical. However, cube root of 16 x squared end root space not equal to space left parenthesis cube root of 16 x end root right parenthesis squared This is because the exponent of 2 applies only to the x, not the 16. (We could rewrite the expression as left parenthesis cube root of 4 x end root right parenthesis squared because 16 = 4 squared)
hint
Taking the odd-root of a negative number leads to a real number solution, because a negative value raised to an odd exponent is negative. However, taking the even-root of a negative value leads to a non-real solution, because a negative value raised to an even exponent is never negative.


3. Applying the Properties of Radicals

When we recognize products, quotients, and powers with radicals, we can apply the properties of radicals to simplify the expression. This is shown in the examples below:

EXAMPLE

cube root of 27 x y cubed end root

cube root of 27 times cube root of x times cube root of x y end root
Product property
3 times cube root of x times y
cube root of 27 equals 3 comma space cube root of y cubed end root equals y
3 y cube root of x
Our Solution

EXAMPLE

fraction numerator cube root of x squared y end root over denominator cube root of 8 x y squared end root end fraction

cube root of fraction numerator x squared y over denominator 8 x y squared end fraction end root

Quotient property
cube root of fraction numerator x over denominator 8 y end fraction end root

One factor of x cancels; one factor of y cancels
cube root of 1 over 8 times x over y end root

Product Property
fraction numerator cube root of 1 over denominator cube root of 8 end fraction times cube root of x y to the power of short dash 1 end exponent end root

Quotient property (andx over y equals x y to the power of short dash 1 end exponent
cube root of 1 over 8 times x over y end root

Quotient property (andx over y equals x y to the power of short dash 1 end exponent
1 half cube root of x y to the power of short dash 1 end exponent end root

Our Solution

summary
We can use the properties of radicals to simplify expressions and solve equations. There are some cautions when applying the properties. The properties of radicals apply only to factors, numbers, and variables combined by multiplication, not by addition or subtraction.

Formulas to Know
Product Property of Radicals

n-th root of a b end root equals n-th root of a times n-th root of b

Property of Fractional Exponents

n-th root of x to the power of m end root equals x to the power of m over n end exponent

Quotient Property of Radicals

n-th root of a over b end root equals fraction numerator n-th root of a over denominator n-th root of b end fraction