3 Tutorials that teach Applying the Properties of Radicals
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Applying the Properties of Radicals

Applying the Properties of Radicals


This lesson applies several properties of radicals in order to simplify expressions.

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  • Properties of Radicals
  • Cautions when Applying the Properties
  • Applying the Properties of Radicals


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

Cautions when Apply the Properties

Avoid these common errors when applying properties of radicals:

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 • square root of 3 because 5 • 3 = 15.  However, we cannot break square root of 8 into square root of 5 space plus space space square root of 3

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 = 42)

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. 

Applying the Properties of Exponents

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: