Solve {left(50^-2right)}^frac{1{2}}{left(5{a}^frac{1{2}}right)}^-2

$${ \left( { 50 }^{ -2 } \right) }^{ \frac{ 1 }{ 2 } } { \left(5 { a }^{ \frac{ 1 }{ 2 } } \right) }^{ -2 }$$

Evaluate

$$\frac{1}{1250a}$$

Solution Steps

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

$$50^{-1}\times \left(5a^{\frac{1}{2}}\right)^{-2}$$

Calculate 50 to the power of -1 and get \frac{1}{50}.

$$\frac{1}{50}\times \left(5a^{\frac{1}{2}}\right)^{-2}$$

Expand \left(5a^{\frac{1}{2}}\right)^{-2}.

$$\frac{1}{50}\times 5^{-2}\left(a^{\frac{1}{2}}\right)^{-2}$$

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

$$\frac{1}{50}\times 5^{-2}a^{-1}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{1}{50}\times \frac{1}{25}a^{-1}$$

Multiply \frac{1}{50} and \frac{1}{25} to get \frac{1}{1250}.

$$\frac{1}{1250}a^{-1}$$

Differentiate w.r.t. a

$$-\frac{1}{1250a^{2}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(50^{-1}\times \left(5a^{\frac{1}{2}}\right)^{-2})$$

Calculate 50 to the power of -1 and get \frac{1}{50}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{50}\times \left(5a^{\frac{1}{2}}\right)^{-2})$$

Expand \left(5a^{\frac{1}{2}}\right)^{-2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{50}\times 5^{-2}\left(a^{\frac{1}{2}}\right)^{-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{50}\times 5^{-2}a^{-1})$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{50}\times \frac{1}{25}a^{-1})$$

Multiply \frac{1}{50} and \frac{1}{25} to get \frac{1}{1250}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{1250}a^{-1})$$

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

$$-\frac{1}{1250}a^{-1-1}$$

Subtract 1 from -1.

$$-\frac{1}{1250}a^{-2}$$