Respuesta :
Answer:
The correct option is D.
Step-by-step explanation:
The given expression is
[tex]\sqrt[3]{7}=7^{\frac{1}{3}}[/tex]
Take cube both sides.
[tex](\sqrt[3]{7})^3=(7^{\frac{1}{3}})^3[/tex]
Using exponent property: [tex]a^3=a\times a\times a[/tex]
[tex](7^{\frac{1}{3}})^3=7^{\frac{1}{3}}\times 7^{\frac{1}{3}}\times 7^{\frac{1}{3}}[/tex]
Using exponent property: [tex]a^ma^n=a^{m+n}[/tex]
[tex](7^{\frac{1}{3}})^3=7^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}[/tex]
[tex](7^{\frac{1}{3}})^3=7^{\frac{3}{3}}[/tex]
[tex](7^{\frac{1}{3}})^3=7^1[/tex]
[tex](7^{\frac{1}{3}})^3=7[/tex]
Therefore option D is correct.
The explanation that correctly explains the considered statement is given by: Option D: [tex](7^{\frac{1}{3}})^3 = 7^{\frac{1}{3}} . 7^{\frac{1}{3}} . 7^{\frac{1}{3}} = 7^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = 7^{\frac{3}{3}} = 7^1 = 7[/tex]
What are some basic properties of exponentiation?
If we have [tex]a^b[/tex] then 'a' is called base and 'b' is called power or exponent and we call it "a is raised to the power b" (this statement might change from text to text slightly).
Exponentiation(the process of raising some number to some power) have some basic rules as:
[tex]a^{-b} = \dfrac{1}{a^b}\\\\a^0 = 1 (a \neq 0)\\\\a^1 = a\\\\(a^b)^c = a^{b \times c}\\\\ a^b \times a^c = a^{b+c} \\\\^n\sqrt{a} = a^{1/n} \\\\(ab)^c = a^c \times b^c[/tex]
For this case, the considered expression is [tex](7^{\frac{1}{3}})^3[/tex]
It can be simplified as:
[tex](7^{\frac{1}{3}})^3 = 7^{\frac{1}{3}} . 7^{\frac{1}{3}} . 7^{\frac{1}{3}}[/tex]
Since bases (all being 7) are same in the right sided multiplication, so the powers(1/3, 1/3, and 1/3) will add up, as shown below:
[tex](7^{\frac{1}{3}})^3 = 7^{\frac{1}{3}} . 7^{\frac{1}{3}} . 7^{\frac{1}{3}} = 7^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = 7^{\frac{3}{3}} = 7^1 = 7[/tex]
Thus, the explanation that correctly explains the considered statement is given by: Option D: [tex](7^{\frac{1}{3}})^3 = 7^{\frac{1}{3}} . 7^{\frac{1}{3}} . 7^{\frac{1}{3}} = 7^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = 7^{\frac{3}{3}} = 7^1 = 7[/tex]
Learn more about exponentiation here:
https://brainly.com/question/847241
