Solve (64x^3div27a^-3)^-frac{2{3}}=?

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

Evaluate

$$\frac{9\times \left(\frac{a}{x}\right)^{2}}{16}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Differentiate w.r.t. x

$$-\frac{9a^{2}}{8x^{3}}$$