To justify why [tex]\( 7^{\frac{1}{3}} = \sqrt[3]{7} \)[/tex], we can use the properties of exponents and roots.
Step-by-Step Explanation:
1. Understanding the notation:
- [tex]\( a^{\frac{1}{n}} \)[/tex] represents the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex].
- In this case, [tex]\( 7^{\frac{1}{3}} \)[/tex] implies the cube root of 7.
2. Exponent and root relationship:
- By definition, [tex]\( a^{\frac{1}{n}} = \sqrt[n]{a} \)[/tex], which means [tex]\( a \)[/tex] raised to the power of [tex]\( \frac{1}{n} \)[/tex] is equivalent to the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex].
3. Application to the specific case:
- Here, [tex]\( a = 7 \)[/tex] and [tex]\( n = 3 \)[/tex].
- Therefore, [tex]\( 7^{\frac{1}{3}} = \sqrt[3]{7} \)[/tex].
So the equation that justifies this relationship is:
[tex]\[ 7^{\frac{1}{3}} = \sqrt[3]{7} \][/tex]
By understanding these properties and relationships, we can confidently say that [tex]\( 7^{\frac{1}{3}} \)[/tex] is indeed equal to [tex]\( \sqrt[3]{7} \)[/tex].
