Solve frac{64*2^{3}*a^8}{10^3*4^3*a^3}

$$\frac { 64 \times 2 ^ { 3 } \times a ^ { 8 } } { 10 ^ { 3 } \times 4 ^ { 3 } \times a ^ { 3 } }$$

Evaluate

$$\frac{a^{5}}{125}$$

Solution Steps

Cancel out a^{3} in both numerator and denominator.

$$\frac{64\times 2^{3}a^{5}}{4^{3}\times 10^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{64\times 8a^{5}}{4^{3}\times 10^{3}}$$

Multiply 64 and 8 to get 512.

$$\frac{512a^{5}}{4^{3}\times 10^{3}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{512a^{5}}{64\times 10^{3}}$$

Calculate 10 to the power of 3 and get 1000.

$$\frac{512a^{5}}{64\times 1000}$$

Multiply 64 and 1000 to get 64000.

$$\frac{512a^{5}}{64000}$$

Divide 512a^{5} by 64000 to get \frac{1}{125}a^{5}.

$$\frac{1}{125}a^{5}$$

Differentiate w.r.t. a

$$\frac{a^{4}}{25}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{512}{64000}a^{8-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{125}a^{5})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$5\times \frac{1}{125}a^{5-1}$$

Do the arithmetic.

$$\frac{1}{25}a^{4}$$