Solve frac{4a^{-1}(a*a^{0}*a)^{-1}}{a^3*a^-4+(a^2)^-frac{1{2}}}

$$\frac { 4 a ^ { - 1 } ( a \cdot a ^ { 0 } \cdot a ) ^ { - 1 } } { a ^ { 3 } \cdot a ^ { - 4 } + ( a ^ { 2 } ) ^ { - \frac { 1 } { 2 } } }$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 0 to get 1.

$$\frac{4a^{-1}\left(a^{1}a\right)^{-1}}{a^{3}a^{-4}+\left(a^{2}\right)^{-\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add 1 and 1 to get 2.

$$\frac{4a^{-1}\left(a^{2}\right)^{-1}}{a^{3}a^{-4}+\left(a^{2}\right)^{-\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{4a^{-1}a^{-2}}{a^{3}a^{-4}+\left(a^{2}\right)^{-\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add -1 and -2 to get -3.

$$\frac{4a^{-3}}{a^{3}a^{-4}+\left(a^{2}\right)^{-\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add 3 and -4 to get -1.

$$\frac{4a^{-3}}{a^{-1}+\left(a^{2}\right)^{-\frac{1}{2}}}$$

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

$$\frac{4a^{-3}}{a^{-1}+a^{-1}}$$

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

$$\frac{4a^{-3}}{2a^{-1}}$$

Cancel out 2 in both numerator and denominator.

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

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

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

Differentiate w.r.t. a

$$-\frac{4}{a^{3}}$$