Solve (8a^{3}div27x^{-3})^frac{2{3}}*(64a^3div27x^-3)^-frac{2{3}}

$$( 8 a ^ { 3 } \div 27 x ^ { - 3 } ) ^ { \frac { 2 } { 3 } } \times ( 64 a ^ { 3 } \div 27 x ^ { - 3 } ) ^ { - \frac { 2 } { 3 } }$$

Evaluate

$$\frac{1}{4}=0.25$$

Solution Steps

Express \frac{8a^{3}}{27}x^{-3} as a single fraction.

$$\left(\frac{8a^{3}x^{-3}}{27}\right)^{\frac{2}{3}}\times \left(\frac{64a^{3}}{27}x^{-3}\right)^{-\frac{2}{3}}$$

To raise \frac{8a^{3}x^{-3}}{27} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8a^{3}x^{-3}\right)^{\frac{2}{3}}}{27^{\frac{2}{3}}}\times \left(\frac{64a^{3}}{27}x^{-3}\right)^{-\frac{2}{3}}$$

Express \frac{64a^{3}}{27}x^{-3} as a single fraction.

$$\frac{\left(8a^{3}x^{-3}\right)^{\frac{2}{3}}}{27^{\frac{2}{3}}}\times \left(\frac{64a^{3}x^{-3}}{27}\right)^{-\frac{2}{3}}$$

To raise \frac{64a^{3}x^{-3}}{27} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8a^{3}x^{-3}\right)^{\frac{2}{3}}}{27^{\frac{2}{3}}}\times \frac{\left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

Expand \left(8a^{3}x^{-3}\right)^{\frac{2}{3}}.

$$\frac{8^{\frac{2}{3}}\left(a^{3}\right)^{\frac{2}{3}}\left(x^{-3}\right)^{\frac{2}{3}}}{27^{\frac{2}{3}}}\times \frac{\left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{2}{3} to get 2.

$$\frac{8^{\frac{2}{3}}a^{2}\left(x^{-3}\right)^{\frac{2}{3}}}{27^{\frac{2}{3}}}\times \frac{\left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and \frac{2}{3} to get -2.

$$\frac{8^{\frac{2}{3}}a^{2}x^{-2}}{27^{\frac{2}{3}}}\times \frac{\left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

Calculate 8 to the power of \frac{2}{3} and get 4.

$$\frac{4a^{2}x^{-2}}{27^{\frac{2}{3}}}\times \frac{\left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

Calculate 27 to the power of \frac{2}{3} and get 9.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{\left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

Expand \left(64a^{3}x^{-3}\right)^{-\frac{2}{3}}.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{64^{-\frac{2}{3}}\left(a^{3}\right)^{-\frac{2}{3}}\left(x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{2}{3} to get -2.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{64^{-\frac{2}{3}}a^{-2}\left(x^{-3}\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -\frac{2}{3} to get 2.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{64^{-\frac{2}{3}}a^{-2}x^{2}}{27^{-\frac{2}{3}}}$$

Calculate 64 to the power of -\frac{2}{3} and get \frac{1}{16}.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{\frac{1}{16}a^{-2}x^{2}}{27^{-\frac{2}{3}}}$$

Calculate 27 to the power of -\frac{2}{3} and get \frac{1}{9}.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{\frac{1}{16}a^{-2}x^{2}}{\frac{1}{9}}$$

Divide \frac{1}{16}a^{-2}x^{2} by \frac{1}{9} by multiplying \frac{1}{16}a^{-2}x^{2} by the reciprocal of \frac{1}{9}.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{1}{16}a^{-2}x^{2}\times 9$$

Multiply \frac{1}{16} and 9 to get \frac{9}{16}.

$$\frac{4a^{2}x^{-2}}{9}\times \frac{9}{16}a^{-2}x^{2}$$

Multiply \frac{4a^{2}x^{-2}}{9} times \frac{9}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4a^{2}x^{-2}\times 9}{9\times 16}a^{-2}x^{2}$$

Cancel out 4\times 9 in both numerator and denominator.

$$\frac{x^{-2}a^{2}}{4}a^{-2}x^{2}$$

Express \frac{x^{-2}a^{2}}{4}a^{-2} as a single fraction.

$$\frac{x^{-2}a^{2}a^{-2}}{4}x^{2}$$

Express \frac{x^{-2}a^{2}a^{-2}}{4}x^{2} as a single fraction.

$$\frac{x^{-2}a^{2}a^{-2}x^{2}}{4}$$

Multiply x^{-2} and x^{2} to get 1.

$$\frac{a^{2}a^{-2}}{4}$$

Multiply a^{2} and a^{-2} to get 1.

$$\frac{1}{4}$$

Factor

$$\frac{1}{2 ^ {2}} = 0.25$$