Solve a^2(-12)2745^-6/15^-330^-8{left(frac{-2{8}right)}^-2}

$$\frac{ { a }^{ 2 } (-12)27 { 45 }^{ -6 } }{ { 15 }^{ -3 } { 30 }^{ -8 } { \left( \frac{ -2 }{ 8 } \right) }^{ -2 } }$$

Evaluate

$$-5400000a^{2}$$

Solution Steps

Multiply -12 and 27 to get -324.

$$\frac{a^{2}\left(-324\right)\times 45^{-6}}{15^{-3}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}}$$

Calculate 45 to the power of -6 and get \frac{1}{8303765625}.

$$\frac{a^{2}\left(-324\right)\times \frac{1}{8303765625}}{15^{-3}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}}$$

Multiply -324 and \frac{1}{8303765625} to get -\frac{4}{102515625}.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{15^{-3}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}}$$

Calculate 15 to the power of -3 and get \frac{1}{3375}.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{3375}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}}$$

Calculate 30 to the power of -8 and get \frac{1}{656100000000}.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{3375}\times \frac{1}{656100000000}\times \left(\frac{-2}{8}\right)^{-2}}$$

Multiply \frac{1}{3375} and \frac{1}{656100000000} to get \frac{1}{2214337500000000}.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{2214337500000000}\times \left(\frac{-2}{8}\right)^{-2}}$$

Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{2214337500000000}\left(-\frac{1}{4}\right)^{-2}}$$

Calculate -\frac{1}{4} to the power of -2 and get 16.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{2214337500000000}\times 16}$$

Multiply \frac{1}{2214337500000000} and 16 to get \frac{1}{138396093750000}.

$$\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{138396093750000}}$$

Divide a^{2}\left(-\frac{4}{102515625}\right) by \frac{1}{138396093750000} by multiplying a^{2}\left(-\frac{4}{102515625}\right) by the reciprocal of \frac{1}{138396093750000}.

$$a^{2}\left(-\frac{4}{102515625}\right)\times 138396093750000$$

Multiply -\frac{4}{102515625} and 138396093750000 to get -5400000.

$$a^{2}\left(-5400000\right)$$

Differentiate w.r.t. a

$$-10800000a$$

Steps Using Definition of a Derivative

Multiply -12 and 27 to get -324.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-324\right)\times 45^{-6}}{15^{-3}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}})$$

Calculate 45 to the power of -6 and get \frac{1}{8303765625}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-324\right)\times \frac{1}{8303765625}}{15^{-3}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}})$$

Multiply -324 and \frac{1}{8303765625} to get -\frac{4}{102515625}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{15^{-3}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}})$$

Calculate 15 to the power of -3 and get \frac{1}{3375}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{3375}\times 30^{-8}\times \left(\frac{-2}{8}\right)^{-2}})$$

Calculate 30 to the power of -8 and get \frac{1}{656100000000}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{3375}\times \frac{1}{656100000000}\times \left(\frac{-2}{8}\right)^{-2}})$$

Multiply \frac{1}{3375} and \frac{1}{656100000000} to get \frac{1}{2214337500000000}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{2214337500000000}\times \left(\frac{-2}{8}\right)^{-2}})$$

Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{2214337500000000}\left(-\frac{1}{4}\right)^{-2}})$$

Calculate -\frac{1}{4} to the power of -2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{2214337500000000}\times 16})$$

Multiply \frac{1}{2214337500000000} and 16 to get \frac{1}{138396093750000}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\left(-\frac{4}{102515625}\right)}{\frac{1}{138396093750000}})$$

Divide a^{2}\left(-\frac{4}{102515625}\right) by \frac{1}{138396093750000} by multiplying a^{2}\left(-\frac{4}{102515625}\right) by the reciprocal of \frac{1}{138396093750000}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}\left(-\frac{4}{102515625}\right)\times 138396093750000)$$

Multiply -\frac{4}{102515625} and 138396093750000 to get -5400000.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}\left(-5400000\right))$$

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

$$2\left(-5400000\right)a^{2-1}$$

Multiply 2 times -5400000.

$$-10800000a^{2-1}$$

Subtract 1 from 2.

$$-10800000a^{1}$$

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

$$-10800000a$$