Write x3/4 in radical form
The Correct Answer and Explanation is :
To express ( x^{3/4} ) in radical form, we can utilize the properties of exponents and radicals. The expression ( x^{3/4} ) can be rewritten as follows:
[
x^{3/4} = \left( x^3 \right)^{1/4}
]
This indicates that we are taking the fourth root of ( x^3 ). In radical notation, this is expressed as:
[
x^{3/4} = \sqrt[4]{x^3}
]
To further clarify this transformation, let’s break it down step by step:
- Understanding Exponents: The fraction ( \frac{3}{4} ) in the exponent can be separated into two parts: the numerator (3) and the denominator (4). The numerator indicates that we will be dealing with ( x^3 ), while the denominator indicates the root we will be taking.
- Applying the Radical: When you have an exponent expressed as ( \frac{m}{n} ), it translates to ( x^m ) being taken to the ( n )-th root. So, for our case, ( m = 3 ) and ( n = 4 ).
- Radical Notation: The notation ( \sqrt[n]{x^m} ) is a way to express roots. Therefore, ( x^{3/4} = \sqrt[4]{x^3} ) fits perfectly with our understanding of exponents and radicals.
- Visualizing the Concept: If ( x ) were a specific number, ( x^{3/4} ) represents a value that is both a power and a root of ( x ). For instance, if ( x = 16 ), then ( x^{3/4} = \sqrt[4]{16^3} = \sqrt[4]{4096} = 6.3496 ) (approximately).
In conclusion, the radical form of ( x^{3/4} ) is ( \sqrt[4]{x^3} ). This transformation highlights the relationship between exponents and roots, providing a different perspective on how to manipulate and understand expressions involving fractional exponents.