Solve (5^3*8a^2)^frac{1{3}}*2^-5

$$( 5 ^ { 3 } \times 8 a ^ { 2 } ) ^ { \frac { 1 } { 3 } } \times 2 ^ { - 5 }$$

Evaluate

$$\frac{5a^{\frac{2}{3}}}{16}$$

Solution Steps

Calculate 5 to the power of 3 and get 125.

$$\left(125\times 8a^{2}\right)^{\frac{1}{3}}\times 2^{-5}$$

Multiply 125 and 8 to get 1000.

$$\left(1000a^{2}\right)^{\frac{1}{3}}\times 2^{-5}$$

Expand \left(1000a^{2}\right)^{\frac{1}{3}}.

$$1000^{\frac{1}{3}}\left(a^{2}\right)^{\frac{1}{3}}\times 2^{-5}$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{1}{3} to get \frac{2}{3}.

$$1000^{\frac{1}{3}}a^{\frac{2}{3}}\times 2^{-5}$$

Calculate 1000 to the power of \frac{1}{3} and get 10.

$$10a^{\frac{2}{3}}\times 2^{-5}$$

Calculate 2 to the power of -5 and get \frac{1}{32}.

$$10a^{\frac{2}{3}}\times \frac{1}{32}$$

Multiply 10 and \frac{1}{32} to get \frac{5}{16}.

$$\frac{5}{16}a^{\frac{2}{3}}$$

Differentiate w.r.t. a

$$\frac{5}{24\sqrt[3]{a}}$$

Steps Using Definition of a Derivative

Calculate 5 to the power of 3 and get 125.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(125\times 8a^{2}\right)^{\frac{1}{3}}\times 2^{-5})$$

Multiply 125 and 8 to get 1000.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(1000a^{2}\right)^{\frac{1}{3}}\times 2^{-5})$$

Expand \left(1000a^{2}\right)^{\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1000^{\frac{1}{3}}\left(a^{2}\right)^{\frac{1}{3}}\times 2^{-5})$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{1}{3} to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1000^{\frac{1}{3}}a^{\frac{2}{3}}\times 2^{-5})$$

Calculate 1000 to the power of \frac{1}{3} and get 10.

$$\frac{\mathrm{d}}{\mathrm{d}a}(10a^{\frac{2}{3}}\times 2^{-5})$$

Calculate 2 to the power of -5 and get \frac{1}{32}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(10a^{\frac{2}{3}}\times \frac{1}{32})$$

Multiply 10 and \frac{1}{32} to get \frac{5}{16}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5}{16}a^{\frac{2}{3}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{2}{3}\times \frac{5}{16}a^{\frac{2}{3}-1}$$

Multiply \frac{2}{3} times \frac{5}{16} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{5}{24}a^{\frac{2}{3}-1}$$

Subtract 1 from \frac{2}{3}.

$$\frac{5}{24}a^{-\frac{1}{3}}$$