To justify why [tex]\( 7^{\frac{1}{3}} = \sqrt[3]{7} \)[/tex], we will use the properties of exponents and radicals. Here is a detailed, step-by-step solution:
1. Understand the Relationship Between Radicals and Exponents:
The expression [tex]\( a^{\frac{1}{n}} \)[/tex] is another way of writing the [tex]\( n \)[/tex]th root of [tex]\( a \)[/tex]. This is a fundamental property of exponents and radicals:
[tex]\[
a^{\frac{1}{n}} = \sqrt[n]{a}
\][/tex]
In this case, [tex]\( a \)[/tex] is 7 and [tex]\( n \)[/tex] is 3. Therefore:
[tex]\[
7^{\frac{1}{3}} = \sqrt[3]{7}
\][/tex]
2. Simplify the Expression:
Let's explicitly consider the value of the expression [tex]\( 7^{\frac{1}{3}} \)[/tex] and the expression [tex]\( \sqrt[3]{7} \)[/tex]. Both are mathematically equivalent and should yield the same numerical result.
[tex]\[
7^{\frac{1}{3}} \approx 1.912931182772389
\][/tex]
[tex]\[
\sqrt[3]{7} \approx 1.912931182772389
\][/tex]
3. Compare the Results:
Since both [tex]\( 7^{\frac{1}{3}} \)[/tex] and [tex]\( \sqrt[3]{7} \)[/tex] result in the same value (approximately 1.912931182772389), we conclude that the two expressions are equal.
[tex]\[
7^{\frac{1}{3}} = \sqrt[3]{7}
\][/tex]
4. Justify the Equation:
The equation that justifies the equality is:
[tex]\[
7^{\frac{1}{3}} = \sqrt[3]{7}
\][/tex]
In conclusion, the expressions [tex]\( 7^{\frac{1}{3}} \)[/tex] and [tex]\( \sqrt[3]{7} \)[/tex] are mathematically equivalent as demonstrated by their identical values. Therefore, the equation that justifies their equality is [tex]\( 7^{\frac{1}{3}} = \sqrt[3]{7} \)[/tex].
