Answer: If m is nonnegative (ie not allowed to be negative), then the answer is m^3 If m is allowed to be negative, then the answer is either |m^3| or |m|^3
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Explanation:
There are two ways to get this answer. The quickest is to simply divide the exponent 6 by 2 to get 6/2 = 3. This value of 3 is the final exponent over the base m. Why do we divide by 2? Because the square root is the same as having an exponent of 1/2 = 0.5, so sqrt(m^6) = (m^6)^(1/2) = m^(6*1/2) = m^(6/2) = m^3 This assumes that m is nonnegative.
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A slightly longer method is to break up the square root into factors of m^2 each and then apply the rule that sqrt(x^2) = x, where x is nonnegative
sqrt(m^6) = sqrt(m^2*m^2*m^2) sqrt(m^6) = sqrt(m^2)*sqrt(m^2)*sqrt(m^2) sqrt(m^6) = m*m*m sqrt(m^6) = m^3 where m is nonnegative
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If we allow m to be negative, then the final result would be either |m^3| or |m|^3.
The reason for the absolute value is to ensure that the expression m^3 is nonnegative. Keep in mind that m^6 is always nonnegative, so sqrt(m^6) is also always nonnegative. In order for sqrt(m^6) = m^3 to be true, the right side must be nonnegative.
Example: Let's say m = -2 m^6 = (-2)^6 = 64 sqrt(m^6) = sqrt(64) = 8 m^3 = (-2)^3 = -8 Without the absolute value, sqrt(m^6) = m^3 is false when m = -2
