To determine which expressions demonstrate the equivalence properties of [tex]\( 4^{\frac{5}{3}} \)[/tex], we will evaluate each expression step-by-step.
1. [tex]\(\sqrt[3]{4^5}=\left(4^5\right)^{\frac{1}{3}}=4^{\frac{5}{3}}\)[/tex]:
[tex]\[
\sqrt[3]{4^5} = (4^5)^{\frac{1}{3}}
\][/tex]
By the properties of exponents, [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(4^5)^{\frac{1}{3}} = 4^{5 \cdot \frac{1}{3}} = 4^{\frac{5}{3}}
\][/tex]
This statement is true.
2. [tex]\(\left(4^{\frac{5}{3}}\right)^3=4^{\left(\frac{5}{3} \cdot 3\right)}=4^5\)[/tex]:
[tex]\[
\left(4^{\frac{5}{3}}\right)^3 = 4^{\frac{5}{3} \cdot 3} = 4^5
\][/tex]
By the exponent multiplication rule:
[tex]\[
4^{\frac{5}{3} \cdot 3} = 4^5
\][/tex]
This statement is true.
3. [tex]\(\left(4^{15}\right)^{\frac{1}{3}}=4^{\left(15 \cdot \frac{1}{3}\right)}=4^5\)[/tex]:
[tex]\[
\left(4^{15}\right)^{\frac{1}{3}} = 4^{15 \cdot \frac{1}{3}} = 4^5
\][/tex]
Simplify the exponent:
[tex]\[
4^{15 \cdot \frac{1}{3}} = 4^{5}
\][/tex]
This may seem correct, but actually, it does not directly relate to [tex]\( 4^{\frac{5}{3}} \)[/tex], hence irrelevant to this context. Thus, we believe it is not necessary.
4. [tex]\(\frac{4^{\frac{17}{3}}}{4^{\frac{2}{3}}}=4^{\left(\frac{17}{3}-\frac{2}{3}\right)}=4^5\)[/tex]:
[tex]\[
\frac{4^{\frac{17}{3}}}{4^{\frac{2}{3}}} = 4^{\frac{17}{3} - \frac{2}{3}} = 4^{\frac{15}{3}} = 4^5
\][/tex]
This statement is true.
5. [tex]\(4^{\frac{8}{3}} \cdot 4^{\frac{7}{3}}=4^{\left(\frac{8}{3}+\frac{7}{3}\right)}=4^5\)[/tex]:
[tex]\[
4^{\frac{8}{3}} \cdot 4^{\frac{7}{3}} = 4^{\frac{8}{3} + \frac{7}{3}} = 4^{\frac{15}{3}} = 4^5
\][/tex]
This statement is true.
In conclusion, the correct answers that show the equivalence properties of [tex]\( 4^{\frac{5}{3}} \)[/tex] are:
- [tex]\(\sqrt[3]{4^5}=\left(4^5\right)^{\frac{1}{3}}=4^{\frac{5}{3}}\)[/tex]
- [tex]\(\left(4^{\frac{5}{3}}\right)^3=4^{\left(\frac{5}{3} \cdot 3\right)}=4^5\)[/tex]
- [tex]\(\frac{4^{\frac{17}{3}}}{4^{\frac{2}{3}}}=4^{\left(\frac{17}{3}-\frac{2}{3}\right)}=4^5\)[/tex]
- [tex]\(4^{\frac{8}{3}} \cdot 4^{\frac{7}{3}}=4^{\left(\frac{9}{3}+\frac{7}{3}\right)}=4^5\)[/tex]
