explain why the square root of a number is defined to be equal to that number to the 1/2 power.

Brief Explanations:

Let's consider the property of exponents ((a^{m})^{n}=a^{mn}). If we assume (y = a^{\frac{1}{2}}), then when we square (y), we get (y^{2}=(a^{\frac{1}{2}})^{2}). Using the exponent rule ((a^{m})^{n}=a^{mn}), here (m = \frac{1}{2}) and (n = 2), so ((a^{\frac{1}{2}})^{2}=a^{\frac{1}{2}\times2}=a^{1}=a). By the definition of a square - root, if (x^{2}=a), then (x=\sqrt{a}). Since (y^{2}=a) where (y = a^{\frac{1}{2}}), the square root of (a) (denoted as (\sqrt{a})) and (a^{\frac{1}{2}}) represent the same value.

Answer:

The square root of a number (a) (denoted (\sqrt{a})) and (a^{\frac{1}{2}}) are equivalent because when we square (a^{\frac{1}{2}}) using the exponent rule ((a^{m})^{n}=a^{mn}) ((m=\frac{1}{2}), (n = 2)), we get (a), which is the same result as when we square the square root of (a) (by the definition of a square root (x^{2}=a) implies (x = \sqrt{a})). So, (\sqrt{a}=a^{\frac{1}{2}}) by the properties of exponents and the definition of a square root.