The redical expressions 3root(2) and 5root(2) are similar.Ģ. Solution root(4,64x^4y^10)= root(4,2^6x^4y^10)ĭEFINITION Radical expressions are said to be similar when they have the same radical index and the same radicand.ĮXAMPLES 1. The cases when there are fractions in the radicand and radicals in the denominator of a fraction will be discussed later.ĮXAMPLE Put root(2^3x^5) in standard form. and the other factors with exponents less than the radical index under the other radical.ĮXAMPLE root(3,x^7)=root(3,x^6*x)=root(3,x^6)root(3,x)=x^2root(3,x) Write the factors with exponents that are integral multiples of the index under one radical, thus obtaining a perfect root.
Then apply the theorem root(n,ab)=root(n,a)root(n,b). When the exponents of some factors of the radicand are greater than the radical index, but not an integral multiple of it, write each of these factors as a product of two factors one factor with an exponent that is an integral multiple of the radical index, and the other factor with an exponent that is less than the radical index. When the radical index and the exponents of all the factors in the radicand have a common factor, divide both the radical index and the exponents of the factors of the radicand by their common factor That is, apply root(nk,a^mk)=root(n,a^m) to obtain the smallest possible radical index. When n is odd and a>0, root(n,-a)= -root(n,a). When n is even and a>0, root(n,-a) is not a real number. When the radicand is negative, the definition gives us the following: The exponent of each factor of the radicand is a natural number less than the radical index.Ĥ. There are no fractions in the radicand.ĥ. There are no radicals in the denominator of a fraction.īy simplifying a radical expression, we mean putting the radical expression in standard form. The radical index is as small as possible.ģ. The expression 3y root(3,x^2y) is called the standard form of the expression root(3,27x^2y^4).Ī radical expression is said to be in standard form if the following conditions hold:Ģ.
Please click "Solve Similar" for more examples.ģ. Let's see some more problems and our step by step solver will simplify the radical expressions. THEOREM If a,b ∈ R, a>0, b>0, and n ∈ N then root(n,ab)=root(n,a)root(n,b). When nis an odd number and a0, a, k>0, we have root(n,m)=root(nk,a^mk), provided nk and mk∈ N. When nis an even number and a0, root(n,a)>0. When n is an even number and a>0, root(n,a)>0, called the principal root. It is a number whose nth power is a that is, (root(n,a))^n with the following conditions:ġ.
For example,ĭEFINITION The nth root of a real number a is denoted by root(n,a). When n is an odd number, the nth power of a positive number is a positive number, and the nth power of a negative number is a negative number. When n is an even number, the nth power of a positive or a negative number is a positive number.
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