UpStudy Free Solution:

To find \(49\) to the power of \(\frac { 1} { 2} \), you are essentially finding the square root of \(49\):

\[49^ { \frac { 1} { 2} } = \sqrt { 49} \]

Since the square root of \(49\) is \(7\):

\[49^ { \frac { 1} { 2} } = 7\]

Supplemental Knowledge

Understanding exponents and radicals is crucial for solving various mathematical problems. Here’s a deeper dive into the concepts:

1. Exponents:

- An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, \(a^ n\) means \(a\) is multiplied by itself \(n\) times.

- Special cases include:

- \(a^ 0 = 1\) (any non-zero number raised to the power of zero is one).

- \(a^ 1 = a\).

2. Fractional Exponents:

- Fractional exponents represent roots. For instance, \(a^ { \frac { 1} { n} } \) is equivalent to the \(n\)-th root of \(a\), written as \(\sqrt [ n] { a} \).

- Example: \(49^ { \frac { 1} { 2} } = \sqrt { 49} \).

3. Square Roots:

- The square root of a number is a value that, when multiplied by itself, gives the original number.

- Notation: \(\sqrt { a} \).

- Example: \(\sqrt { 49} = 7\), because \(7 \times 7 = 49\).

4. Properties of Exponents and Radicals:

- Product Rule: \(( a \cdot b) ^ n = a^ n \cdot b^ n\).

- Quotient Rule: \(( \frac { a} { b} ) ^ n = \frac { a^ n} { b^ n} \).

- Power Rule: \(( a^ m) ^ n = a^ { m \cdot n} \).

5. Applications:

- These principles are widely used in algebra, calculus, physics, engineering, and many other fields.

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