Solve (3a^{2}b^{7})^{3}:(a^{2}b^2)*(3a^2b^3)^5

$$( 3 a ^ { 2 } b ^ { 7 } ) ^ { 3 } : ( a ^ { 2 } b ^ { 2 } ) \cdot ( 3 a ^ { 2 } b ^ { 3 } ) ^ { 5 }$$

Evaluate

$$6561a^{14}b^{34}$$

Solution Steps

Expand \left(3a^{2}b^{7}\right)^{3}.

$$\frac{3^{3}\left(a^{2}\right)^{3}\left(b^{7}\right)^{3}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{3^{3}a^{6}\left(b^{7}\right)^{3}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$\frac{3^{3}a^{6}b^{21}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{27a^{6}b^{21}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

Cancel out a^{2}b^{2} in both numerator and denominator.

$$27a^{4}b^{19}\times \left(3a^{2}b^{3}\right)^{5}$$

Expand \left(3a^{2}b^{3}\right)^{5}.

$$27a^{4}b^{19}\times 3^{5}\left(a^{2}\right)^{5}\left(b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$27a^{4}b^{19}\times 3^{5}a^{10}\left(b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$27a^{4}b^{19}\times 3^{5}a^{10}b^{15}$$

Calculate 3 to the power of 5 and get 243.

$$27a^{4}b^{19}\times 243a^{10}b^{15}$$

Multiply 27 and 243 to get 6561.

$$6561a^{4}b^{19}a^{10}b^{15}$$

To multiply powers of the same base, add their exponents. Add 4 and 10 to get 14.

$$6561a^{14}b^{19}b^{15}$$

To multiply powers of the same base, add their exponents. Add 19 and 15 to get 34.

$$6561a^{14}b^{34}$$

Expand

$$6561a^{14}b^{34}$$

Solution Steps

Expand \left(3a^{2}b^{7}\right)^{3}.

$$\frac{3^{3}\left(a^{2}\right)^{3}\left(b^{7}\right)^{3}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{3^{3}a^{6}\left(b^{7}\right)^{3}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$\frac{3^{3}a^{6}b^{21}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{27a^{6}b^{21}}{a^{2}b^{2}}\times \left(3a^{2}b^{3}\right)^{5}$$

Cancel out a^{2}b^{2} in both numerator and denominator.

$$27a^{4}b^{19}\times \left(3a^{2}b^{3}\right)^{5}$$

Expand \left(3a^{2}b^{3}\right)^{5}.

$$27a^{4}b^{19}\times 3^{5}\left(a^{2}\right)^{5}\left(b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$27a^{4}b^{19}\times 3^{5}a^{10}\left(b^{3}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$27a^{4}b^{19}\times 3^{5}a^{10}b^{15}$$

Calculate 3 to the power of 5 and get 243.

$$27a^{4}b^{19}\times 243a^{10}b^{15}$$

Multiply 27 and 243 to get 6561.

$$6561a^{4}b^{19}a^{10}b^{15}$$

To multiply powers of the same base, add their exponents. Add 4 and 10 to get 14.

$$6561a^{14}b^{19}b^{15}$$

To multiply powers of the same base, add their exponents. Add 19 and 15 to get 34.

$$6561a^{14}b^{34}$$