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To simplify an n^(th) root, the radicand must first be expressed as a power. If the index of the radical and the power of the radicand are equal such that sqrt(a^n), the radical expression can be simplified as follows.
sqrt(a^n)= a, & if n is odd |a|, & if n is even
The absolute value of a number is always non-negative, so when n is even, the result will always be non-negative. Consider a few example n^\text{th} roots that can be simplified by using the formula.
a | n | sqrt(a^n) | Is n even? | Simplify |
---|---|---|---|---|
4 | 3 | sqrt(4^3) | * | 4 |
-6 | 5 | sqrt((-6)^5) | * | -6 |
2 | 6 | sqrt(2^6) | âś“ | |2| = 2 |
-3 | 4 | sqrt((-3)^4) | âś“ | |-3| = 3 |
To write the expression for sqrt(a^n), there are two cases to consider.
Both cases will be considered one at a time.
In case of a negative value of a, there are also two cases two consider.
(- a)^(n_e) = a^(n_e)
sqrt(a)=a^(1n)
(a^m)^n=a^(m* n)
a * 1/a=1
a^1=a
A root with an odd index n_o is defined for all real numbers. By the definition of the n^\text{th} root, the expression sqrt(a^(n_o)) is the number y that, when multiplied by itself n_o times, will result in a^(n_o). y * y * ... * y_(n_otimes)=a^(n_o) Because n_o is odd and a is negative, a^(n_o) is also negative. This means that the best candidate for y is simply a. sqrt(a^n) = a
If a is non-negative, sqrt(a^n) is always equal to a. However, in case of negative a, the value of sqrt(a^n) depends on the parity of n.
a≥ 0 | a<0 | |
---|---|---|
Even n | sqrt(a^n) = a | sqrt(a^n) = - a |
Odd n | sqrt(a^n) = a | sqrt(a^n) = a |
To conclude, for odd values of n, the expression sqrt(a^n) is equal to a. On the other hand, if n is even, sqrt(a^n) can be written as |a|.
sqrt(a^n)= a, & if n is odd |a|, & if n is even
Depending on the index of the root and the power in the radicand, simplifying sqrt(a^m) may be problematic. Because real n^\text{th} roots with an even index are defined only for non-negative numbers, the absolute value is sometimes needed.
If n is even, sqrt(a^m) is defined only for non-negative a^m.
a | m | n | sqrt(a^m) | Simplify |
---|---|---|---|---|
2 | 8 | 4 | sqrt(2^8) | 2^()84 = 2^2 = 4 |
-3 | 6 | 2 | sqrt((-3)^6) | | (-3)^()62 | = | (-3)^3 | = |-27| = 27 |
x | 6 | 3 | sqrt(x^6) | x^()63 = x^2 |
-3 | 2 | 8 | sqrt((-3)^2) | sqrt(|-3|) = sqrt(3) |
2 | 3 | 6 | sqrt(2^3) | sqrt(2) |
-2 | 3 | 9 | sqrt((-2)^3) | sqrt(-2) |
2 | 6 | 8 | sqrt(2^6) | sqrt(2^3) = sqrt(8) |
-2 | 6 | 8 | sqrt((-2)^6) | sqrt(|-2|^3) = sqrt(2^3) = sqrt(8) |
Given two non-negative numbers a and b, the n^\text{th} root of their product equals the product of the n^\text{th} root of each number.
sqrt(ab) = sqrt(a)* sqrt(b), for a≥ 0 and b≥ 0
If n is an odd number, the n^\text{th} root of a negative number is defined. In this case, the Product Property of Radicals for negative a and b is also true.
y^n= b
ab= z^n
a^m b^m = (a b)^m
Rearrange equation
Let a be a non-negative number and b be a positive number. The n^\text{th} root of the quotient ab equals the quotient of the n^\text{th} roots of a and b.
sqrt(a/b) = sqrt(a)/sqrt(b), for a≥ 0, b > 0
If n is an odd number, the n^\text{th} root of a negative number is defined. In this case, the Quotient Property of Radicals for negative a and b is also true.
y^n= b
a/b= z^n
a^m/b^m=(a/b)^m
Rearrange equation
Let a be a non-negative number and b be a positive number. The n^\text{th} root of the quotient ab equals the quotient of the n^\text{th} roots of a and b.
sqrt(a/b) = sqrt(a)/sqrt(b), for a≥ 0, b > 0
If n is an odd number, the n^\text{th} root of a negative number is defined. In this case, the Quotient Property of Radicals for negative a and b is also true.
y^n= b
a/b= z^n
a^m/b^m=(a/b)^m
Rearrange equation
Factor the number in the radicand into its prime factors and the variables into their smallest exponents. In this example, the number under the radical is 50 and it is a product of powers of 2 and 5. 50 = 2 * 5^2 Since x^6 and y^4 are already prime factors, the radical expression can be rewritten as follows. sqrt(50x^6y^4) = sqrt(2 * 5^2 * x^6 * y^4)
Identify and separate perfect powers that match the index of the radical. Identifying perfect squares for a square root means recognizing expressions that can be rewritten as the product of two identical powers with integer exponents. In the given example, 5^2, x^6, and y^4 are all perfect squares. sqrt(2 * 5^2 * x^6 * y^4) = sqrt(2 * 5^2 * (x^3)^2 * (y^2)^2)
sqrt(a* b)=sqrt(a)*sqrt(b)
sqrt(a^2)=a
sqrt(a^2)=|a|
Finally, combine the simplified terms outside the root while keeping any terms that are not perfect powers inside the radical. sqrt(2) * 5 * |x^3| * y^2 = 5sqrt(2)|x^3| y^2