Why is $\sqrt[3]{9}$ equal to $9^{\frac{1}{3}}$?
Answer (1)
The n -th root of a number a can be expressed as a raised to the power of n 1 .
Express the relationship between radicals and fractional exponents: n a = a n 1 .
Apply the definition to the specific case: 3 9 = 9 3 1 .
Conclude that 3 9 is equal to 9 3 1 based on the definition. \boxed{\sqrt[3]{9} = 9^{\frac{1}{3}}}$ ### Explanation 1. Understanding the Problem The question asks us to explain why the cube root of 9, denoted as $\sqrt[3]{9}$, is equal to 9 raised to the power of $\frac{1}{3}$, denoted as $9^{\frac{1}{3}}$. The provided calculations are not relevant to the explanation. 2. Recalling the Definition The fundamental relationship between radicals and fractional exponents states that the $n$-th root of a number $a$ can be expressed as $a$ raised to the power of $\frac{1}{n}$. Mathematically, this is written as: \sqrt[n]{a} = a^{\frac{1}{n}} This is a definition and a property that holds true for all positive real numbers $a$ and positive integers $n$. 3. Applying the Definition to the Problem Applying this general rule to our specific case, where $a = 9$ and $n = 3$, we directly get: \sqrt[3]{9} = 9^{\frac{1}{3}} This equation tells us that the cube root of 9 is, by definition, equal to 9 raised to the power of 3 1 .
Conclusion Therefore, 3 9 is equal to 9 3 1 because this is the definition of fractional exponents and radicals.
Examples
Imagine you're dividing a square garden into equal parts. Taking the square root is like finding the side length of a smaller square that perfectly fits within the original. Similarly, a cube root helps you find the side of a cube that fits perfectly inside a larger cube. Fractional exponents are just a way to represent these roots, making calculations easier. For example, if you have a garden with an area of 9 m 2 and want to divide it into four equal square plots, you would use the square root to find the side length of each smaller plot. This concept is useful in various fields, from construction to design, where understanding proportions and scaling is essential.
