Solve (36x)^frac{1{2}}*(27x^6)^frac{1{3}}

$$( 36 x ) ^ { \frac { 1 } { 2 } } \cdot ( 27 x ^ { 6 } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$18x^{\frac{5}{2}}$$

Solution Steps

Expand \left(36x\right)^{\frac{1}{2}}.

$$36^{\frac{1}{2}}x^{\frac{1}{2}}\times \left(27x^{6}\right)^{\frac{1}{3}}$$

Calculate 36 to the power of \frac{1}{2} and get 6.

$$6x^{\frac{1}{2}}\times \left(27x^{6}\right)^{\frac{1}{3}}$$

Expand \left(27x^{6}\right)^{\frac{1}{3}}.

$$6x^{\frac{1}{2}}\times 27^{\frac{1}{3}}\left(x^{6}\right)^{\frac{1}{3}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{3} to get 2.

$$6x^{\frac{1}{2}}\times 27^{\frac{1}{3}}x^{2}$$

Calculate 27 to the power of \frac{1}{3} and get 3.

$$6x^{\frac{1}{2}}\times 3x^{2}$$

Multiply 6 and 3 to get 18.

$$18x^{\frac{1}{2}}x^{2}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 2 to get \frac{5}{2}.

$$18x^{\frac{5}{2}}$$

Differentiate w.r.t. x

$$45x^{\frac{3}{2}}$$

Steps Using Definition of a Derivative

Expand \left(36x\right)^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(36^{\frac{1}{2}}x^{\frac{1}{2}}\times \left(27x^{6}\right)^{\frac{1}{3}})$$

Calculate 36 to the power of \frac{1}{2} and get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{\frac{1}{2}}\times \left(27x^{6}\right)^{\frac{1}{3}})$$

Expand \left(27x^{6}\right)^{\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{\frac{1}{2}}\times 27^{\frac{1}{3}}\left(x^{6}\right)^{\frac{1}{3}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{3} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{\frac{1}{2}}\times 27^{\frac{1}{3}}x^{2})$$

Calculate 27 to the power of \frac{1}{3} and get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{\frac{1}{2}}\times 3x^{2})$$

Multiply 6 and 3 to get 18.

$$\frac{\mathrm{d}}{\mathrm{d}x}(18x^{\frac{1}{2}}x^{2})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 2 to get \frac{5}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(18x^{\frac{5}{2}})$$

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

$$\frac{5}{2}\times 18x^{\frac{5}{2}-1}$$

Multiply \frac{5}{2} times 18.

$$45x^{\frac{5}{2}-1}$$

Subtract 1 from \frac{5}{2}.

$$45x^{\frac{3}{2}}$$