Solve (3a^{2})^{3}:(-1/3a)^{5}-a(a^{12}:a^7)+(-9)^4a+(-5a^3)^2

$$( 3 a ^ { 2 } ) ^ { 3 } : ( - \frac { 1 } { 3 } a ) ^ { 5 } - a ( a ^ { 12 } : a ^ { 7 } ) + ( - 9 ) ^ { 4 } a + ( - 5 a ^ { 3 } ) ^ { 2 }$$

Evaluate

$$24a^{6}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 7 from 12 to get 5.

$$\frac{\left(3a^{2}\right)^{3}}{\left(-\frac{1}{3}a\right)^{5}}-aa^{5}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$\frac{\left(3a^{2}\right)^{3}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

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

$$\frac{3^{3}\left(a^{2}\right)^{3}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

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

$$\frac{3^{3}a^{6}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{27a^{6}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Expand \left(-\frac{1}{3}a\right)^{5}.

$$\frac{27a^{6}}{\left(-\frac{1}{3}\right)^{5}a^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Calculate -\frac{1}{3} to the power of 5 and get -\frac{1}{243}.

$$\frac{27a^{6}}{-\frac{1}{243}a^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Cancel out a^{5} in both numerator and denominator.

$$\frac{27a}{-\frac{1}{243}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Divide 27a by -\frac{1}{243} by multiplying 27a by the reciprocal of -\frac{1}{243}.

$$\frac{27a\times 243}{-1}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Multiply 27 and 243 to get 6561.

$$\frac{6561a}{-1}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Anything divided by -1 gives its opposite.

$$-6561a-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Calculate -9 to the power of 4 and get 6561.

$$-6561a-a^{6}+6561a+\left(-5a^{3}\right)^{2}$$

Combine -6561a and 6561a to get 0.

$$-a^{6}+\left(-5a^{3}\right)^{2}$$

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

$$-a^{6}+\left(-5\right)^{2}\left(a^{3}\right)^{2}$$

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

$$-a^{6}+\left(-5\right)^{2}a^{6}$$

Calculate -5 to the power of 2 and get 25.

$$-a^{6}+25a^{6}$$

Combine -a^{6} and 25a^{6} to get 24a^{6}.

$$24a^{6}$$

Expand

$$24a^{6}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 7 from 12 to get 5.

$$\frac{\left(3a^{2}\right)^{3}}{\left(-\frac{1}{3}a\right)^{5}}-aa^{5}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$\frac{\left(3a^{2}\right)^{3}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

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

$$\frac{3^{3}\left(a^{2}\right)^{3}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

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

$$\frac{3^{3}a^{6}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{27a^{6}}{\left(-\frac{1}{3}a\right)^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Expand \left(-\frac{1}{3}a\right)^{5}.

$$\frac{27a^{6}}{\left(-\frac{1}{3}\right)^{5}a^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Calculate -\frac{1}{3} to the power of 5 and get -\frac{1}{243}.

$$\frac{27a^{6}}{-\frac{1}{243}a^{5}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Cancel out a^{5} in both numerator and denominator.

$$\frac{27a}{-\frac{1}{243}}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Divide 27a by -\frac{1}{243} by multiplying 27a by the reciprocal of -\frac{1}{243}.

$$\frac{27a\times 243}{-1}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Multiply 27 and 243 to get 6561.

$$\frac{6561a}{-1}-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Anything divided by -1 gives its opposite.

$$-6561a-a^{6}+\left(-9\right)^{4}a+\left(-5a^{3}\right)^{2}$$

Calculate -9 to the power of 4 and get 6561.

$$-6561a-a^{6}+6561a+\left(-5a^{3}\right)^{2}$$

Combine -6561a and 6561a to get 0.

$$-a^{6}+\left(-5a^{3}\right)^{2}$$

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

$$-a^{6}+\left(-5\right)^{2}\left(a^{3}\right)^{2}$$

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

$$-a^{6}+\left(-5\right)^{2}a^{6}$$

Calculate -5 to the power of 2 and get 25.

$$-a^{6}+25a^{6}$$

Combine -a^{6} and 25a^{6} to get 24a^{6}.

$$24a^{6}$$