Solve {left(frac{frac{8sqrt{2}}{3}}{2}right)}^2+{left(frac{frac{8sqrt{3}}{3}}{2}right)}^2=16

$${ \left( \frac{ \frac{ 8 \sqrt{ 2 } }{ 3 } }{ 2 } \right) }^{ 2 } + { \left( \frac{ \frac{ 8 \sqrt{ 3 } }{ 3 } }{ 2 } \right) }^{ 2 } =16$$

Verify

Solution Steps

Express \frac{\frac{8\sqrt{2}}{3}}{2} as a single fraction.

$$\left(\frac{8\sqrt{2}}{3\times 2}\right)^{2}+\left(\frac{\frac{8\sqrt{3}}{3}}{2}\right)^{2}=16$$

Cancel out 2 in both numerator and denominator.

$$\left(\frac{4\sqrt{2}}{3}\right)^{2}+\left(\frac{\frac{8\sqrt{3}}{3}}{2}\right)^{2}=16$$

To raise \frac{4\sqrt{2}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(4\sqrt{2}\right)^{2}}{3^{2}}+\left(\frac{\frac{8\sqrt{3}}{3}}{2}\right)^{2}=16$$

Express \frac{\frac{8\sqrt{3}}{3}}{2} as a single fraction.

$$\frac{\left(4\sqrt{2}\right)^{2}}{3^{2}}+\left(\frac{8\sqrt{3}}{3\times 2}\right)^{2}=16$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(4\sqrt{2}\right)^{2}}{3^{2}}+\left(\frac{4\sqrt{3}}{3}\right)^{2}=16$$

To raise \frac{4\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(4\sqrt{2}\right)^{2}}{3^{2}}+\frac{\left(4\sqrt{3}\right)^{2}}{3^{2}}=16$$

Since \frac{\left(4\sqrt{2}\right)^{2}}{3^{2}} and \frac{\left(4\sqrt{3}\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4\sqrt{2}\right)^{2}+\left(4\sqrt{3}\right)^{2}}{3^{2}}=16$$

Expand \left(4\sqrt{2}\right)^{2}.

$$\frac{4^{2}\left(\sqrt{2}\right)^{2}+\left(4\sqrt{3}\right)^{2}}{3^{2}}=16$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16\left(\sqrt{2}\right)^{2}+\left(4\sqrt{3}\right)^{2}}{3^{2}}=16$$

The square of \sqrt{2} is 2.

$$\frac{16\times 2+\left(4\sqrt{3}\right)^{2}}{3^{2}}=16$$

Multiply 16 and 2 to get 32.

$$\frac{32+\left(4\sqrt{3}\right)^{2}}{3^{2}}=16$$

Expand \left(4\sqrt{3}\right)^{2}.

$$\frac{32+4^{2}\left(\sqrt{3}\right)^{2}}{3^{2}}=16$$

Calculate 4 to the power of 2 and get 16.

$$\frac{32+16\left(\sqrt{3}\right)^{2}}{3^{2}}=16$$

The square of \sqrt{3} is 3.

$$\frac{32+16\times 3}{3^{2}}=16$$

Multiply 16 and 3 to get 48.

$$\frac{32+48}{3^{2}}=16$$

Add 32 and 48 to get 80.

$$\frac{80}{3^{2}}=16$$

Calculate 3 to the power of 2 and get 9.

$$\frac{80}{9}=16$$

Convert 16 to fraction \frac{144}{9}.

$$\frac{80}{9}=\frac{144}{9}$$

Compare \frac{80}{9} and \frac{144}{9}.

$$\text{false}$$