What is the square root of m6?
The correct answer and explanation is:
Sure! To clarify, I assume you mean the square root of m⁶ (m raised to the 6th power), not “m6” as a number or term.
Correct Answer:
m6=m62=m3\sqrt{m^6} = m^{\frac{6}{2}} = m^3
Explanation (300 words):
When taking the square root of an expression involving exponents, it is important to remember the relationship between roots and powers.
The square root of any expression xx can be rewritten as an exponent of one-half: x=x1/2\sqrt{x} = x^{1/2}
So, applying this to m6m^6: m6=(m6)1/2\sqrt{m^6} = (m^6)^{1/2}
When you raise a power to another power, you multiply the exponents: (m6)1/2=m6×12=m3(m^6)^{1/2} = m^{6 \times \frac{1}{2}} = m^3
This simplification is valid for any real number mm, assuming mm is non-negative if working in the realm of real numbers because the square root function for real numbers is typically defined for non-negative values.
Why does this work?
Exponent rules dictate that: (am)n=am×n(a^m)^n = a^{m \times n}
So, taking the square root is essentially raising to the power of 12\frac{1}{2}, and the power inside (6) gets multiplied by 12\frac{1}{2}, giving the exponent 3.
Practical example:
If m=2m = 2: 26=64=8\sqrt{2^6} = \sqrt{64} = 8
and 23=82^3 = 8
Both methods give the same result, confirming the rule.
Summary:
- Square root = power of 1/2.
- When taking a root of a power, multiply exponents.
- So, m6=m3\sqrt{m^6} = m^3.