Answer :
To determine which property provides for the definition of \( x^{\frac{a}{b}} \) as \( \sqrt[b]{x^a} \), let's understand the different properties related to exponents:
### Examining the Properties
1. Power of a Power Property:
- This property states that \((x^a)^b = x^{a \cdot b}\).
- Applying this idea in reverse tells us that raising a number to a fraction means we are looking for a root. For example, \( x^{1/2} \) means the square root of \( x \), and \( x^{1/3} \) means the cube root of \( x \).
2. Product of Powers Property:
- This property applies when you multiply two exponents with the same base: \( x^a \cdot x^b = x^{a+b} \).
- While important, it doesn't directly inform us about fractional exponents.
3. Quotient of Powers Property:
- This property is used when dividing two exponents with the same base: \( \frac{x^a}{x^b} = x^{a-b} \).
- Again, while useful, it doesn't directly address the concept of fractional exponents.
4. Square Root Property:
- This property specifically refers to the square root of a number: if \( y = \sqrt{x} \), then \( y^2 = x \).
- It does not generalize to other roots or fractional exponents.
### Application to the Problem
Given the definition \( x^{\frac{a}{b}} = \sqrt[b]{x^a} \), we can see this specifically involves raising something to a power and then taking the root. This matches the description of the Power of a Power Property because it demonstrates how exponents (powers) can be manipulated to achieve a simplified understanding, especially concerning fractional exponents and roots.
Thus, the correct answer is:
A. Power of a Power Property
### Examining the Properties
1. Power of a Power Property:
- This property states that \((x^a)^b = x^{a \cdot b}\).
- Applying this idea in reverse tells us that raising a number to a fraction means we are looking for a root. For example, \( x^{1/2} \) means the square root of \( x \), and \( x^{1/3} \) means the cube root of \( x \).
2. Product of Powers Property:
- This property applies when you multiply two exponents with the same base: \( x^a \cdot x^b = x^{a+b} \).
- While important, it doesn't directly inform us about fractional exponents.
3. Quotient of Powers Property:
- This property is used when dividing two exponents with the same base: \( \frac{x^a}{x^b} = x^{a-b} \).
- Again, while useful, it doesn't directly address the concept of fractional exponents.
4. Square Root Property:
- This property specifically refers to the square root of a number: if \( y = \sqrt{x} \), then \( y^2 = x \).
- It does not generalize to other roots or fractional exponents.
### Application to the Problem
Given the definition \( x^{\frac{a}{b}} = \sqrt[b]{x^a} \), we can see this specifically involves raising something to a power and then taking the root. This matches the description of the Power of a Power Property because it demonstrates how exponents (powers) can be manipulated to achieve a simplified understanding, especially concerning fractional exponents and roots.
Thus, the correct answer is:
A. Power of a Power Property
