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

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

Evaluate

$$\frac{1}{4a^{3}}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of -2 and get \frac{1}{4}.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Multiply \frac{1}{4} and 4 to get 1.

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

To multiply powers of the same base, add their exponents. Add -4 and -6 to get -10.

$$\frac{a^{-10}}{\left(2^{2}a^{-1}\right)^{3}\times \left(2a\right)^{-4}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{a^{-10}}{\left(4a^{-1}\right)^{3}\times \left(2a\right)^{-4}}$$

Expand \left(4a^{-1}\right)^{3}.

$$\frac{a^{-10}}{4^{3}\left(a^{-1}\right)^{3}\times \left(2a\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{a^{-10}}{4^{3}a^{-3}\times \left(2a\right)^{-4}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{a^{-10}}{64a^{-3}\times \left(2a\right)^{-4}}$$

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

$$\frac{a^{-10}}{64a^{-3}\times 2^{-4}a^{-4}}$$

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$\frac{a^{-10}}{64a^{-3}\times \frac{1}{16}a^{-4}}$$

Multiply 64 and \frac{1}{16} to get 4.

$$\frac{a^{-10}}{4a^{-3}a^{-4}}$$

To multiply powers of the same base, add their exponents. Add -3 and -4 to get -7.

$$\frac{a^{-10}}{4a^{-7}}$$

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

$$\frac{1}{4a^{3}}$$

Expand

$$\frac{1}{4a^{3}}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of -2 and get \frac{1}{4}.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Multiply \frac{1}{4} and 4 to get 1.

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

To multiply powers of the same base, add their exponents. Add -4 and -6 to get -10.

$$\frac{a^{-10}}{\left(2^{2}a^{-1}\right)^{3}\times \left(2a\right)^{-4}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{a^{-10}}{\left(4a^{-1}\right)^{3}\times \left(2a\right)^{-4}}$$

Expand \left(4a^{-1}\right)^{3}.

$$\frac{a^{-10}}{4^{3}\left(a^{-1}\right)^{3}\times \left(2a\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{a^{-10}}{4^{3}a^{-3}\times \left(2a\right)^{-4}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{a^{-10}}{64a^{-3}\times \left(2a\right)^{-4}}$$

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

$$\frac{a^{-10}}{64a^{-3}\times 2^{-4}a^{-4}}$$

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$\frac{a^{-10}}{64a^{-3}\times \frac{1}{16}a^{-4}}$$

Multiply 64 and \frac{1}{16} to get 4.

$$\frac{a^{-10}}{4a^{-3}a^{-4}}$$

To multiply powers of the same base, add their exponents. Add -3 and -4 to get -7.

$$\frac{a^{-10}}{4a^{-7}}$$

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

$$\frac{1}{4a^{3}}$$