Solve frac{(-7)^{-2}*3^{-1}*11^{-2}*a^-4}{22^-4*21^-3*a^-6}

$$\frac { ( - 7 ) ^ { - 2 } \times 3 ^ { - 1 } \times 11 ^ { - 2 } \times a ^ { - 4 } } { 22 ^ { - 4 } \times 21 ^ { - 3 } \times a ^ { - 6 } }$$

Evaluate

$$121968a^{2}$$

Solution Steps

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

$$\frac{\left(-7\right)^{-2}\times 11^{-2}\times \frac{1}{3}a^{2}}{21^{-3}\times 22^{-4}}$$

Calculate -7 to the power of -2 and get \frac{1}{49}.

$$\frac{\frac{1}{49}\times 11^{-2}\times \frac{1}{3}a^{2}}{21^{-3}\times 22^{-4}}$$

Calculate 11 to the power of -2 and get \frac{1}{121}.

$$\frac{\frac{1}{49}\times \frac{1}{121}\times \frac{1}{3}a^{2}}{21^{-3}\times 22^{-4}}$$

Multiply \frac{1}{49} and \frac{1}{121} to get \frac{1}{5929}.

$$\frac{\frac{1}{5929}\times \frac{1}{3}a^{2}}{21^{-3}\times 22^{-4}}$$

Multiply \frac{1}{5929} and \frac{1}{3} to get \frac{1}{17787}.

$$\frac{\frac{1}{17787}a^{2}}{21^{-3}\times 22^{-4}}$$

Calculate 21 to the power of -3 and get \frac{1}{9261}.

$$\frac{\frac{1}{17787}a^{2}}{\frac{1}{9261}\times 22^{-4}}$$

Calculate 22 to the power of -4 and get \frac{1}{234256}.

$$\frac{\frac{1}{17787}a^{2}}{\frac{1}{9261}\times \frac{1}{234256}}$$

Multiply \frac{1}{9261} and \frac{1}{234256} to get \frac{1}{2169444816}.

$$\frac{\frac{1}{17787}a^{2}}{\frac{1}{2169444816}}$$

Divide \frac{1}{17787}a^{2} by \frac{1}{2169444816} by multiplying \frac{1}{17787}a^{2} by the reciprocal of \frac{1}{2169444816}.

$$\frac{1}{17787}a^{2}\times 2169444816$$

Multiply \frac{1}{17787} and 2169444816 to get 121968.

$$121968a^{2}$$

Differentiate w.r.t. a

$$243936a$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{1}{17787}}{\frac{1}{2169444816}}a^{-4-\left(-6\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(121968a^{2})$$

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}.

$$2\times 121968a^{2-1}$$

Do the arithmetic.

$$243936a^{1}$$

For any term t, t^{1}=t.

$$243936a$$