Solve frac{8{a}^frac{2{4}}{a}^frac{1{4}}2{b}^frac{1{2}}}{4{b}^frac{1{2}}}

$$\frac{ 8 { a }^{ \frac{ 2 }{ 4 } } { a }^{ \frac{ 1 }{ 4 } } 2 { b }^{ \frac{ 1 }{ 2 } } }{ 4 { b }^{ \frac{ 1 }{ 2 } } }$$

Evaluate

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

Solution Steps

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{8a^{\frac{1}{2}}a^{\frac{1}{4}}\times 2b^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{4} to get \frac{3}{4}.

$$\frac{8a^{\frac{3}{4}}\times 2b^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$$

Multiply 8 and 2 to get 16.

$$\frac{16a^{\frac{3}{4}}b^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$$

Cancel out 4\sqrt{b} in both numerator and denominator.

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

Differentiate w.r.t. a

$$\frac{3}{\sqrt[4]{a}}$$

Steps Using Definition of a Derivative

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{8a^{\frac{1}{2}}a^{\frac{1}{4}}\times 2b^{\frac{1}{2}}}{4b^{\frac{1}{2}}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{4} to get \frac{3}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{8a^{\frac{3}{4}}\times 2b^{\frac{1}{2}}}{4b^{\frac{1}{2}}})$$

Multiply 8 and 2 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{16a^{\frac{3}{4}}b^{\frac{1}{2}}}{4b^{\frac{1}{2}}})$$

Cancel out 4\sqrt{b} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(4a^{\frac{3}{4}})$$

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

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

Multiply \frac{3}{4} times 4.

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

Subtract 1 from \frac{3}{4}.

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