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what's covered

Properties of Radicals
Cautions when Applying the Properties
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

. However, we cannot break square root of 8

into

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

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

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

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


		Product property

		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
		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 (and
		Quotient property (and
		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: $n-th root of a over b end root equals fraction numerator n-th root of a over denominator n-th root of b end fraction$

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1. Properties of Radicals

2. Cautions when Apply the Properties

3. Applying the Properties of Radicals