Dive into the world of knowledge and get your queries resolved at IDNLearn.com. Join our knowledgeable community and access a wealth of reliable answers to your most pressing questions.
Sagot :
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]
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]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.