Solve frac{(2h)^{2}*(16h^8)^frac{1{4}}}{(8^frac{1{3}}h)^-2}

$$\frac { ( 2 h ) ^ { 2 } \times ( 16 h ^ { 8 } ) ^ { \frac { 1 } { 4 } } } { ( 8 ^ { \frac { 1 } { 3 } } h ) ^ { - 2 } }$$

Evaluate

$$32h^{6}$$

Solution Steps

Expand \left(2h\right)^{2}.

$$\frac{2^{2}h^{2}\times \left(16h^{8}\right)^{\frac{1}{4}}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4h^{2}\times \left(16h^{8}\right)^{\frac{1}{4}}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

Expand \left(16h^{8}\right)^{\frac{1}{4}}.

$$\frac{4h^{2}\times 16^{\frac{1}{4}}\left(h^{8}\right)^{\frac{1}{4}}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and \frac{1}{4} to get 2.

$$\frac{4h^{2}\times 16^{\frac{1}{4}}h^{2}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

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

$$\frac{4h^{2}\times 2h^{2}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

Multiply 4 and 2 to get 8.

$$\frac{8h^{2}h^{2}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{8h^{4}}{\left(8^{\frac{1}{3}}h\right)^{-2}}$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{8h^{4}}{\left(2h\right)^{-2}}$$

Expand \left(2h\right)^{-2}.

$$\frac{8h^{4}}{2^{-2}h^{-2}}$$

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

$$\frac{8h^{4}}{\frac{1}{4}h^{-2}}$$

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

$$\frac{8h^{6}}{\frac{1}{4}}$$

Divide 8h^{6} by \frac{1}{4} by multiplying 8h^{6} by the reciprocal of \frac{1}{4}.

$$8h^{6}\times 4$$

Multiply 8 and 4 to get 32.

$$32h^{6}$$

Differentiate w.r.t. h

$$192h^{5}$$

Steps Using Definition of a Derivative

Expand \left(2h\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{2^{2}h^{2}\times \left(16h^{8}\right)^{\frac{1}{4}}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{4h^{2}\times \left(16h^{8}\right)^{\frac{1}{4}}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

Expand \left(16h^{8}\right)^{\frac{1}{4}}.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{4h^{2}\times 16^{\frac{1}{4}}\left(h^{8}\right)^{\frac{1}{4}}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

To raise a power to another power, multiply the exponents. Multiply 8 and \frac{1}{4} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{4h^{2}\times 16^{\frac{1}{4}}h^{2}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{4h^{2}\times 2h^{2}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{8h^{2}h^{2}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{8h^{4}}{\left(8^{\frac{1}{3}}h\right)^{-2}})$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{8h^{4}}{\left(2h\right)^{-2}})$$

Expand \left(2h\right)^{-2}.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{8h^{4}}{2^{-2}h^{-2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{8h^{4}}{\frac{1}{4}h^{-2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{8h^{6}}{\frac{1}{4}})$$

Divide 8h^{6} by \frac{1}{4} by multiplying 8h^{6} by the reciprocal of \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}h}(8h^{6}\times 4)$$

Multiply 8 and 4 to get 32.

$$\frac{\mathrm{d}}{\mathrm{d}h}(32h^{6})$$

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

$$6\times 32h^{6-1}$$

Multiply 6 times 32.

$$192h^{6-1}$$

Subtract 1 from 6.

$$192h^{5}$$