Solve frac{(3a^{2})^{5}*(9a^{4})^{2}*(b^{7})^{2}}{(81a^2)^3*(b^2)^7}

$$\frac { ( 3 a ^ { 2 } ) ^ { 5 } \cdot ( 9 a ^ { 4 } ) ^ { 2 } \cdot ( b ^ { 7 } ) ^ { 2 } } { ( 81 a ^ { 2 } ) ^ { 3 } \cdot ( b ^ { 2 } ) ^ { 7 } }$$

Evaluate

$$\frac{a^{12}}{27}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{\left(3a^{2}\right)^{5}\times \left(9a^{4}\right)^{2}b^{14}}{\left(81a^{2}\right)^{3}\left(b^{2}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 7 to get 14.

$$\frac{\left(3a^{2}\right)^{5}\times \left(9a^{4}\right)^{2}b^{14}}{\left(81a^{2}\right)^{3}b^{14}}$$

Cancel out b^{14} in both numerator and denominator.

$$\frac{\left(9a^{4}\right)^{2}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

Expand \left(9a^{4}\right)^{2}.

$$\frac{9^{2}\left(a^{4}\right)^{2}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{9^{2}a^{8}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{81a^{8}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

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

$$\frac{81a^{8}\times 3^{5}\left(a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

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

$$\frac{81a^{8}\times 3^{5}a^{10}}{\left(81a^{2}\right)^{3}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{81a^{8}\times 243a^{10}}{\left(81a^{2}\right)^{3}}$$

Multiply 81 and 243 to get 19683.

$$\frac{19683a^{8}a^{10}}{\left(81a^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 8 and 10 to get 18.

$$\frac{19683a^{18}}{\left(81a^{2}\right)^{3}}$$

Expand \left(81a^{2}\right)^{3}.

$$\frac{19683a^{18}}{81^{3}\left(a^{2}\right)^{3}}$$

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

$$\frac{19683a^{18}}{81^{3}a^{6}}$$

Calculate 81 to the power of 3 and get 531441.

$$\frac{19683a^{18}}{531441a^{6}}$$

Cancel out 19683a^{6} in both numerator and denominator.

$$\frac{a^{12}}{27}$$

Expand

$$\frac{a^{12}}{27}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{\left(3a^{2}\right)^{5}\times \left(9a^{4}\right)^{2}b^{14}}{\left(81a^{2}\right)^{3}\left(b^{2}\right)^{7}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 7 to get 14.

$$\frac{\left(3a^{2}\right)^{5}\times \left(9a^{4}\right)^{2}b^{14}}{\left(81a^{2}\right)^{3}b^{14}}$$

Cancel out b^{14} in both numerator and denominator.

$$\frac{\left(9a^{4}\right)^{2}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

Expand \left(9a^{4}\right)^{2}.

$$\frac{9^{2}\left(a^{4}\right)^{2}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{9^{2}a^{8}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{81a^{8}\times \left(3a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

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

$$\frac{81a^{8}\times 3^{5}\left(a^{2}\right)^{5}}{\left(81a^{2}\right)^{3}}$$

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

$$\frac{81a^{8}\times 3^{5}a^{10}}{\left(81a^{2}\right)^{3}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{81a^{8}\times 243a^{10}}{\left(81a^{2}\right)^{3}}$$

Multiply 81 and 243 to get 19683.

$$\frac{19683a^{8}a^{10}}{\left(81a^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 8 and 10 to get 18.

$$\frac{19683a^{18}}{\left(81a^{2}\right)^{3}}$$

Expand \left(81a^{2}\right)^{3}.

$$\frac{19683a^{18}}{81^{3}\left(a^{2}\right)^{3}}$$

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

$$\frac{19683a^{18}}{81^{3}a^{6}}$$

Calculate 81 to the power of 3 and get 531441.

$$\frac{19683a^{18}}{531441a^{6}}$$

Cancel out 19683a^{6} in both numerator and denominator.

$$\frac{a^{12}}{27}$$