Solve (a^{3}*b^{3})^frac{1{2}}*(a^frac{1{2}}*b^3)^4

$$( a ^ { 3 } \cdot b ^ { 3 } ) ^ { \frac { 1 } { 2 } } \cdot ( a ^ { \frac { 1 } { 2 } } \cdot b ^ { 3 } ) ^ { 4 }$$

Evaluate

$$\left(a^{\frac{7}{3}}b^{9}\right)^{\frac{3}{2}}$$

Solution Steps

Expand \left(a^{3}b^{3}\right)^{\frac{1}{2}}.

$$\left(a^{3}\right)^{\frac{1}{2}}\left(b^{3}\right)^{\frac{1}{2}}\left(a^{\frac{1}{2}}b^{3}\right)^{4}$$

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

$$a^{\frac{3}{2}}\left(b^{3}\right)^{\frac{1}{2}}\left(a^{\frac{1}{2}}b^{3}\right)^{4}$$

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

$$a^{\frac{3}{2}}b^{\frac{3}{2}}\left(a^{\frac{1}{2}}b^{3}\right)^{4}$$

Expand \left(a^{\frac{1}{2}}b^{3}\right)^{4}.

$$a^{\frac{3}{2}}b^{\frac{3}{2}}\left(a^{\frac{1}{2}}\right)^{4}\left(b^{3}\right)^{4}$$

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

$$a^{\frac{3}{2}}b^{\frac{3}{2}}a^{2}\left(b^{3}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$a^{\frac{3}{2}}b^{\frac{3}{2}}a^{2}b^{12}$$

To multiply powers of the same base, add their exponents. Add \frac{3}{2} and 2 to get \frac{7}{2}.

$$a^{\frac{7}{2}}b^{\frac{3}{2}}b^{12}$$

To multiply powers of the same base, add their exponents. Add \frac{3}{2} and 12 to get \frac{27}{2}.

$$a^{\frac{7}{2}}b^{\frac{27}{2}}$$

Differentiate w.r.t. b

$$\frac{27a^{\frac{7}{3}}\sqrt{a^{\frac{7}{3}}b^{25}}}{2}$$