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10.1  Deﬁnitions and Notation

The nth powers of 2,a,3^2, and b^3 are, respectively, 22^n,a^n,3^(2n), and b^(3n).
When n is an even number, the nth power of a positive or a negative number is a positive number. For example,

(02)^4=16  and  (-2)^4=16

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. For example,

(+3)^3=27  and  (-3)^3=-27.

DEFINITION  The nth root of a real number a is denoted by root(n,a). It is a number whose nth power is a; that is, (root(n,a))^n with the following conditions:

1. When n is an even number and a>0,root(n,a)>0, called the principal root.

When nis an even number and  a<0,root(n,a) is not a real number.

2. When nis an odd number and  a>0,root(n,a)>0.

When nis an odd number and  a<0,root(n,a)<0

The n in root(n,a) (always a natural number greater than 1) is called the index or the order of the radical, and a is called the radicand. When there is no indicated index, as in root(a), the index 2 is implied and it is read “the square root of a." When the index is 3 as in root(3,a), it is read “the cube root of a."

EXAMPLES  1. root(49)=root(7^2)=(root(7))^2=7

2. -root(25)=-root((5)^2)=-(root(5))^2=-5

3. root(-4) is not a real number

4. root(3,8)=root(3,(2)^3)=(root(2))^3=2

5. root(5,-32)=root(5,(-2)^5=(root(5,-2)^5=-2=-root(5,32)

6. root(4,x^8=root(4,(x^2)^4)=(root(4,2))^4=x^2

7. root(5,x^15)=root(5,(x^3)^5=(root(5,x^3))^5=x^3

From the deﬁnition of fractional exponents (page 321) and the deﬁnition of radicals, for a {is-in} {real}m,n {is-in} {natural}, we have

root(n,a)=a^(1/n)  and  root(n,a^m)=a^(m/n)

providing root(n,a) and a^(1/n) are defined.

The above relations enable us to express radicals as fractional exponents and fractional exponents as radicals.

EXAMPLES  1. root(5,3)=3^(1/5)

2. root(3,2^2)=2^(2/3)

3. root(x+3)=(x+3)^(1/2)

4. x^(4/5)=root(5,x^4)

5. 3x^(3/4)=3root(4,x^3)

6. x^(1/2)y^(2/3)=root(x)root(3,y^2)

When the value of a radical expression is a rational number, we say it is a perfect root Since root(n,a^nk)=a^k, a radical expression is a perfect root if the radicand can be expressed as a product of factors each to an exponent that is an integral multiple of the radical index.

The value of the radical is obtained by forming the product of the factors. where the exponent of each factor is its original exponent divided by the radical index.

EXAMPLES  1.  root(5^6)=5^(6/2)=5^3

2.  root(x^10)=x^(10/2)=x^5

3.  root(3,8x^6y^9=root(3,2^3x^6y^9=2^(3/3)x^(6/3)y^(9/3)=2x^2y^3

Note  Nonperfect roots such as root(2),root(3,2),root(3),root(4,5),root(5,4),1+root(2) and 5-root(3,9) are irrational numbers. An irrational number is a number that cannot be expressed in the form p/q, where p,q{Iota},q!=0.

Note  1. Since a^(m/n)=a^(mk/nk) for all a>0a {is-in} {real}, and m,nNk {is-in} {rat}k>0, we have root(n,m)=root(nk,a^mk), provided nk and mkN.

root(3,a)=root(6,a^2)  and  root(n,1)=1

2. 1^n=1  and  root(n,1)=1.

THEOREM   If a,bRa>0,b>0, and nN then root(n,ab)=root(n,a)root(n,b).

Proof     root(n,ab)=(ab)^(1/n)=a^(1/n)b^(1/n)=root(n,a)root(n,b)

EXAMPLES  1. root(32)=root(2^5)=root(2^4*2^1)

=root(2^4)*root(2)

=2^2root(2)=4root(2)

2. root(16x^3y=root(2^4x^3y=root(2^4x^2xy)

=root(2^4x^2)root(xy)

=2^2xroot(xy)

=4xroot(xy)

Let's see some more problems and our step by step solver will simplify the radical expressions. Please click "Solve Similar" for more examples.

3. root(3,27x^2y^4)=root(3,3^3x^2y^4)=root(3,3^3z^2y^3y

=root(3,3^3y^3)root(3,x^2y)

=3yroot(3,x^2y)

The expression 3yroot(3,x^2y) is called the standard form of the expression root(3,27x^2y^4).

A radical expression is said to be in standard form if the following conditions hold:

2. The radical index is as small as possible.

3. The exponent of each factor of the radicand is a natural number less than the radical index.

4. There are no fractions in the radicand.

5. There are no radicals in the denominator of a fraction.

By simplifying a radical expression, we mean putting the radical expression in standard form.

When the radicand is negative, the deﬁnition gives us the following:

When n is even and a>0,  root(n,-a) is not a real number.

When n is odd and a>0,  root(n,-a)=-root(n,a).

EXAMPLES  1. root(3,-5)=-root(3,5)

2. root(5,-x^2y^3)=-root(5,x^2y^3)

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.

EXAMPLE    root(6,a^2b^4)=root(3,ab^2)

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. For example,

root(2,x^7)=root(3,x^6*x)

Then apply the theorem root(n,ab)=root(n,a)root(n,b). Write the factors with exponents that are integral multiples of the index under one radical, thus obtaining a perfect root. and the other factors with exponents less than the radical index under the other radical.

EXAMPLE    root(3,x^7)=root(3,x^6*x)=root(3,x^6)root(3,x)=x^2root(3,x)

The cases when there are fractions in the radicand and radicals in the denominator of a fraction will be discussed later.

EXAMPLE    Put root(2^3x^5) in standard form.

Solution   root(2^3x^5)=root((2^2*2)(x^4*x)

=root(2^2x^4)root(2x)

=2x^2root(2x)

EXAMPLE    Put root(8x^3y^2z^5) in standard form.

Solution   root(8x^3y^2z^5)=root(2^3x^3y^2z^5)

=root((2^2*2)(x^2*x)y^2(z^4.z)

=root(2^2x^2y^2z^4)root(2xz)

=2xyz^2root(2xz)

EXAMPLE    Put root(3,2^4x^6y^5z^10) in standard form.

Solution   root(3,2^4x^6y^5z^10)=root(3,(2^3*2)x^6(y^3*y^2)(z^9*z)

root(3,2^3x^6y^3z^9)root(3,2y^2z)

=2x^2yz^3root(3,2y^2z)

EXAMPLE    Put root(3,-2x^11y^4in standard form.

Solution    root(3,-2x^11y^4=-root(3,2(x^9*x^2)(y^3*y))

=-root(3,(x^9y^3))root(3,2x^2y)

=-x^3yroot(3,2x^2y)

EXAMPLE    Put root(4,64x^4y^10) in standard form.

Solution    root(4,64x^4y^10)=root(4,2^6x^4y^10)

=root(4,(2^4*2^2)x^4(y^8*y^2))

=root(4,2^4x^4y^8)root(4,2^2y^2)

=2xy^2root(2y)

DEFINITION  Radical expressions are said to be similar when they have the same radical index and the same radicand.

EXAMPLES  1. The redical expressions 3root(2) and 5root(2) are similar.

2. The redical expressions root(24) and root(54) can be shown to be similar.

root(24)=root(2^3*3)=root(2^2*2*3=2root(2)

and root(54)=root(2*3^3)=root(2*3^2*3=3root(6)

3. The redical expressions rootroot(18) and root(27) are not similar.

root(18)=root(2*3^2)=3root(2)

and root(27)=root(3^3)=root(3^2*3)=3root(3).

Radical expressions can be combined only when they are similar. First we put the radical expressions in standard form and then combine similar radicals using the distributive law.

EXAMPLE  Simplify root(54)-root(24)+root(150) and combine similar redical expressions.

Solution   root(54)-root(24)+root(150)

=root(2*3^3)-root(2^3*3)+root(2*3*5^2)

=3root(6)-2root(6)+5root(6)

=(3-2+5)root(6)

=6root(6)

EXAMPLE  Simplify xroot(147y^3)+yroot(75x^2y)-root(48x^2y^3) and combine similar redicals.

Solution   xroot(147y^3)+yroot(75x^2y)-root(48x^2y^3)

=xroot(3*7^2y^3)+yroot(3*5^2x^2y)-root(2^4*3x^2y^3)

=7xyroot(3y)+5xyroot(3y)-4xyroot(3y)

=8xyroot(3y)

Let's see some more problems and our step by step solver will simplify the combination of radical expressions. Please click "Solve Similar" for more examples.

EXAMPLE  Simplify 3root(8)-root(3,81)-root(128)+root(3,375) and combine similar redicals .

Solution    3root(8)-root(3,81)-root(128)+root(3,375)

=3root(2^3)-root(3,3^4)-root(2^7)root(3,3*5^3)

=3*2root(2)-3root(3,3)-2^3root(2)+5root(3,3)

=6root(2)-3root(3,3)-8root(2)+5root(3,3)

=(6-8)root(2)+(-3+5)root(3,3)

=-2root(2)+2root(3,3)