Solve [(-1/4a^{2}bc)^{2}*3c^{2}]:(-1/2a^{2}b+3/4a^{2}b)^{2}+(-1/2ab^{2}c^{3})^2:(-frac{1}{2}a^2b^4c^2)

$$[ ( - \frac { 1 } { 4 } a ^ { 2 } b c ) ^ { 2 } \cdot 3 c ^ { 2 } ] : ( - \frac { 1 } { 2 } a ^ { 2 } b + \frac { 3 } { 4 } a ^ { 2 } b ) ^ { 2 } + ( - \frac { 1 } { 2 } a b ^ { 2 } c ^ { 3 } ) ^ { 2 } : ( - \frac { 1 } { 2 } a ^ { 2 } b ^ { 4 } c ^ { 2 } )$$

Evaluate

$$\frac{5c^{4}}{2}$$

Solution Steps

Expand \left(-\frac{1}{4}a^{2}bc\right)^{2}.

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

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

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

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

$$\frac{\frac{1}{16}a^{4}b^{2}c^{2}\times 3c^{2}}{\left(-\frac{1}{2}a^{2}b+\frac{3}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Multiply \frac{1}{16} and 3 to get \frac{3}{16}.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{2}c^{2}}{\left(-\frac{1}{2}a^{2}b+\frac{3}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(-\frac{1}{2}a^{2}b+\frac{3}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Combine -\frac{1}{2}a^{2}b and \frac{3}{4}a^{2}b to get \frac{1}{4}a^{2}b.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(\frac{1}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Expand \left(\frac{1}{4}a^{2}b\right)^{2}.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(\frac{1}{4}\right)^{2}\left(a^{2}\right)^{2}b^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(\frac{1}{4}\right)^{2}a^{4}b^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\frac{1}{16}a^{4}b^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Cancel out b^{2}a^{4} in both numerator and denominator.

$$\frac{\frac{3}{16}c^{4}}{\frac{1}{16}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Divide \frac{3}{16}c^{4} by \frac{1}{16} by multiplying \frac{3}{16}c^{4} by the reciprocal of \frac{1}{16}.

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Expand \left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}.

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}\right)^{2}a^{2}\left(b^{2}\right)^{2}\left(c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}\right)^{2}a^{2}b^{4}\left(c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}\right)^{2}a^{2}b^{4}c^{6}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{4}a^{2}b^{4}c^{6}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Cancel out a^{2}c^{2}b^{4} in both numerator and denominator.

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{4}c^{4}}{-\frac{1}{2}}$$

Divide \frac{1}{4}c^{4} by -\frac{1}{2} by multiplying \frac{1}{4}c^{4} by the reciprocal of -\frac{1}{2}.

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{4}c^{4}\times 2}{-1}$$

Multiply \frac{1}{4} and 2 to get \frac{1}{2}.

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{2}c^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$\frac{3}{16}c^{4}\times 16-\frac{1}{2}c^{4}$$

Multiply \frac{3}{16} and 16 to get 3.

$$3c^{4}-\frac{1}{2}c^{4}$$

Combine 3c^{4} and -\frac{1}{2}c^{4} to get \frac{5}{2}c^{4}.

$$\frac{5}{2}c^{4}$$

Expand

$$\frac{5c^{4}}{2}$$

Solution Steps

Expand \left(-\frac{1}{4}a^{2}bc\right)^{2}.

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

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

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

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

$$\frac{\frac{1}{16}a^{4}b^{2}c^{2}\times 3c^{2}}{\left(-\frac{1}{2}a^{2}b+\frac{3}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Multiply \frac{1}{16} and 3 to get \frac{3}{16}.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{2}c^{2}}{\left(-\frac{1}{2}a^{2}b+\frac{3}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(-\frac{1}{2}a^{2}b+\frac{3}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Combine -\frac{1}{2}a^{2}b and \frac{3}{4}a^{2}b to get \frac{1}{4}a^{2}b.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(\frac{1}{4}a^{2}b\right)^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Expand \left(\frac{1}{4}a^{2}b\right)^{2}.

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(\frac{1}{4}\right)^{2}\left(a^{2}\right)^{2}b^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\left(\frac{1}{4}\right)^{2}a^{4}b^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{\frac{3}{16}a^{4}b^{2}c^{4}}{\frac{1}{16}a^{4}b^{2}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Cancel out b^{2}a^{4} in both numerator and denominator.

$$\frac{\frac{3}{16}c^{4}}{\frac{1}{16}}+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Divide \frac{3}{16}c^{4} by \frac{1}{16} by multiplying \frac{3}{16}c^{4} by the reciprocal of \frac{1}{16}.

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Expand \left(-\frac{1}{2}ab^{2}c^{3}\right)^{2}.

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}\right)^{2}a^{2}\left(b^{2}\right)^{2}\left(c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}\right)^{2}a^{2}b^{4}\left(c^{3}\right)^{2}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{3}{16}c^{4}\times 16+\frac{\left(-\frac{1}{2}\right)^{2}a^{2}b^{4}c^{6}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

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

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{4}a^{2}b^{4}c^{6}}{-\frac{1}{2}a^{2}b^{4}c^{2}}$$

Cancel out a^{2}c^{2}b^{4} in both numerator and denominator.

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{4}c^{4}}{-\frac{1}{2}}$$

Divide \frac{1}{4}c^{4} by -\frac{1}{2} by multiplying \frac{1}{4}c^{4} by the reciprocal of -\frac{1}{2}.

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{4}c^{4}\times 2}{-1}$$

Multiply \frac{1}{4} and 2 to get \frac{1}{2}.

$$\frac{3}{16}c^{4}\times 16+\frac{\frac{1}{2}c^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$\frac{3}{16}c^{4}\times 16-\frac{1}{2}c^{4}$$

Multiply \frac{3}{16} and 16 to get 3.

$$3c^{4}-\frac{1}{2}c^{4}$$

Combine 3c^{4} and -\frac{1}{2}c^{4} to get \frac{5}{2}c^{4}.

$$\frac{5}{2}c^{4}$$