Question

the surface area of a cubical shipping box with a volume of ( v ) cubic units is described by the expression below. ( 6v^{\frac{2}{3}} ) determine whether or not each expression given below is equivalent to the expression for the surface area. drag and drop \equivalent\ or
ot equivalent\ into the boxes to match each expression. expression equivalent or not equivalent to the above expression for surface area? ( sqrt3{6v^2} ) ( 6sqrt3{v^2} ) ( 6sqrt2{v^8} ) ( 6sqrt4{v^5} ) ( 6sqrt{v}sqrt3{v} ) ( \frac{6v}{sqrt3{v}} )

Explanation:

To determine if each expression is equivalent to \(6V^{\frac{2}{3}}\), we use the property of exponents: \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) and \(\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}\), and \( \frac{a^m}{a^n}=a^{m - n}\).

1. For \(\sqrt[3]{6V^2}\)

Using the property \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\), we have \(\sqrt[3]{6V^2}=\sqrt[3]{6}\cdot\sqrt[3]{V^2}=\sqrt[3]{6}V^{\frac{2}{3}}
eq6V^{\frac{2}{3}}\) (since \(\sqrt[3]{6}
eq6\)). So it is Not Equivalent.

2. For \(6\sqrt[3]{V^2}\)

Using the property \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), we know that \(\sqrt[3]{V^2}=V^{\frac{2}{3}}\). So \(6\sqrt[3]{V^2} = 6V^{\frac{2}{3}}\). Thus, it is Equivalent.

3. For \(6\sqrt[12]{V^8}\)

First, simplify the exponent of \(V\) in the radical. Using the property \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), we have \(\sqrt[12]{V^8}=V^{\frac{8}{12}}=V^{\frac{2}{3}}\). So \(6\sqrt[12]{V^8}=6V^{\frac{2}{3}}\). Thus, it is Equivalent.

4. For \(6\sqrt[4]{V^5}\)

Simplify the exponent of \(V\): \(\sqrt[4]{V^5}=V^{\frac{5}{4}}\). Then \(6\sqrt[4]{V^5}=6V^{\frac{5}{4}}
eq6V^{\frac{2}{3}}\) (since \(\frac{5}{4}
eq\frac{2}{3}\)). So it is Not Equivalent.

5. For \(6\sqrt{V}\sqrt[3]{V}\)

First, rewrite the radicals as exponents: \(\sqrt{V}=V^{\frac{1}{2}}\) and \(\sqrt[3]{V}=V^{\frac{1}{3}}\). Then, using the property \(a^m\cdot a^n=a^{m + n}\), we have \(V^{\frac{1}{2}}\cdot V^{\frac{1}{3}}=V^{\frac{1}{2}+\frac{1}{3}}=V^{\frac{3 + 2}{6}}=V^{\frac{5}{6}}\). So \(6\sqrt{V}\sqrt[3]{V}=6V^{\frac{5}{6}}
eq6V^{\frac{2}{3}}\) (since \(\frac{5}{6}
eq\frac{2}{3}\)). Thus, it is Not Equivalent.

6. For \(\frac{6V}{\sqrt[3]{V}}\)

Using the property \(\frac{a^m}{a^n}=a^{m - n}\), we have \(\frac{V}{\sqrt[3]{V}}=\frac{V^1}{V^{\frac{1}{3}}}=V^{1-\frac{1}{3}}=V^{\frac{2}{3}}\). So \(\frac{6V}{\sqrt[3]{V}}=6V^{\frac{2}{3}}\). Thus, it is Equivalent.

Final Answers:

• \(\sqrt[3]{6V^2}\): Not Equivalent
• \(6\sqrt[3]{V^2}\): Equivalent
• \(6\sqrt[12]{V^8}\): Equivalent
• \(6\sqrt[4]{V^5}\): Not Equivalent
• \(6\sqrt{V}\sqrt[3]{V}\): Not Equivalent
• \(\frac{6V}{\sqrt[3]{V}}\): Equivalent

Answer:

To determine if each expression is equivalent to \(6V^{\frac{2}{3}}\), we use the property of exponents: \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) and \(\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}\), and \( \frac{a^m}{a^n}=a^{m - n}\).

1. For \(\sqrt[3]{6V^2}\)

Using the property \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\), we have \(\sqrt[3]{6V^2}=\sqrt[3]{6}\cdot\sqrt[3]{V^2}=\sqrt[3]{6}V^{\frac{2}{3}}
eq6V^{\frac{2}{3}}\) (since \(\sqrt[3]{6}
eq6\)). So it is Not Equivalent.

2. For \(6\sqrt[3]{V^2}\)

Using the property \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), we know that \(\sqrt[3]{V^2}=V^{\frac{2}{3}}\). So \(6\sqrt[3]{V^2} = 6V^{\frac{2}{3}}\). Thus, it is Equivalent.

3. For \(6\sqrt[12]{V^8}\)

First, simplify the exponent of \(V\) in the radical. Using the property \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), we have \(\sqrt[12]{V^8}=V^{\frac{8}{12}}=V^{\frac{2}{3}}\). So \(6\sqrt[12]{V^8}=6V^{\frac{2}{3}}\). Thus, it is Equivalent.

4. For \(6\sqrt[4]{V^5}\)

Simplify the exponent of \(V\): \(\sqrt[4]{V^5}=V^{\frac{5}{4}}\). Then \(6\sqrt[4]{V^5}=6V^{\frac{5}{4}}
eq6V^{\frac{2}{3}}\) (since \(\frac{5}{4}
eq\frac{2}{3}\)). So it is Not Equivalent.

5. For \(6\sqrt{V}\sqrt[3]{V}\)

First, rewrite the radicals as exponents: \(\sqrt{V}=V^{\frac{1}{2}}\) and \(\sqrt[3]{V}=V^{\frac{1}{3}}\). Then, using the property \(a^m\cdot a^n=a^{m + n}\), we have \(V^{\frac{1}{2}}\cdot V^{\frac{1}{3}}=V^{\frac{1}{2}+\frac{1}{3}}=V^{\frac{3 + 2}{6}}=V^{\frac{5}{6}}\). So \(6\sqrt{V}\sqrt[3]{V}=6V^{\frac{5}{6}}
eq6V^{\frac{2}{3}}\) (since \(\frac{5}{6}
eq\frac{2}{3}\)). Thus, it is Not Equivalent.

6. For \(\frac{6V}{\sqrt[3]{V}}\)

Using the property \(\frac{a^m}{a^n}=a^{m - n}\), we have \(\frac{V}{\sqrt[3]{V}}=\frac{V^1}{V^{\frac{1}{3}}}=V^{1-\frac{1}{3}}=V^{\frac{2}{3}}\). So \(\frac{6V}{\sqrt[3]{V}}=6V^{\frac{2}{3}}\). Thus, it is Equivalent.

Final Answers:

• \(\sqrt[3]{6V^2}\): Not Equivalent
• \(6\sqrt[3]{V^2}\): Equivalent
• \(6\sqrt[12]{V^8}\): Equivalent
• \(6\sqrt[4]{V^5}\): Not Equivalent
• \(6\sqrt{V}\sqrt[3]{V}\): Not Equivalent
• \(\frac{6V}{\sqrt[3]{V}}\): Equivalent