Solve frac{[a^{2}*a^{-3}*(1/a)^3]^2}{(1/a)^-2*a^2}

$$\frac { [ a ^ { 2 } \cdot a ^ { - 3 } \cdot ( \frac { 1 } { a } ) ^ { 3 } ] ^ { 2 } } { ( \frac { 1 } { a } ) ^ { - 2 } \cdot a ^ { 2 } }$$

Evaluate

$$\frac{1}{a^{12}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and -3 to get -1.

$$\frac{\left(a^{-1}\times \left(\frac{1}{a}\right)^{3}\right)^{2}}{\left(\frac{1}{a}\right)^{-2}a^{2}}$$

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{-1}\times \frac{1^{3}}{a^{3}}\right)^{2}}{\left(\frac{1}{a}\right)^{-2}a^{2}}$$

Express a^{-1}\times \frac{1^{3}}{a^{3}} as a single fraction.

$$\frac{\left(\frac{a^{-1}\times 1^{3}}{a^{3}}\right)^{2}}{\left(\frac{1}{a}\right)^{-2}a^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

To raise \frac{1^{3}}{a^{4}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

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

Express \frac{1^{-2}}{a^{-2}}a^{2} as a single fraction.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

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

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

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

$$\frac{\frac{1^{6}}{a^{8}}}{1^{-2}a^{4}}$$

Calculate 1 to the power of 6 and get 1.

$$\frac{\frac{1}{a^{8}}}{1^{-2}a^{4}}$$

Calculate 1 to the power of -2 and get 1.

$$\frac{\frac{1}{a^{8}}}{1a^{4}}$$

Express \frac{\frac{1}{a^{8}}}{1a^{4}} as a single fraction.

$$\frac{1}{a^{8}\times 1a^{4}}$$

To multiply powers of the same base, add their exponents. Add 8 and 4 to get 12.

$$\frac{1}{a^{12}\times 1}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{a^{12}}$$

Expand

$$\frac{1}{a^{12}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and -3 to get -1.

$$\frac{\left(a^{-1}\times \left(\frac{1}{a}\right)^{3}\right)^{2}}{\left(\frac{1}{a}\right)^{-2}a^{2}}$$

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{-1}\times \frac{1^{3}}{a^{3}}\right)^{2}}{\left(\frac{1}{a}\right)^{-2}a^{2}}$$

Express a^{-1}\times \frac{1^{3}}{a^{3}} as a single fraction.

$$\frac{\left(\frac{a^{-1}\times 1^{3}}{a^{3}}\right)^{2}}{\left(\frac{1}{a}\right)^{-2}a^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

To raise \frac{1^{3}}{a^{4}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

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

Express \frac{1^{-2}}{a^{-2}}a^{2} as a single fraction.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

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

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

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

$$\frac{\frac{1^{6}}{a^{8}}}{1^{-2}a^{4}}$$

Calculate 1 to the power of 6 and get 1.

$$\frac{\frac{1}{a^{8}}}{1^{-2}a^{4}}$$

Calculate 1 to the power of -2 and get 1.

$$\frac{\frac{1}{a^{8}}}{1a^{4}}$$

Express \frac{\frac{1}{a^{8}}}{1a^{4}} as a single fraction.

$$\frac{1}{a^{8}\times 1a^{4}}$$

To multiply powers of the same base, add their exponents. Add 8 and 4 to get 12.

$$\frac{1}{a^{12}\times 1}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{a^{12}}$$