Solve (-2/3a^{5}b^{4}d^{3})^{2}:(-8/15a^{2}b^2d)^2:(-15/2a^2bd)^2

$$( - \frac { 2 } { 3 } a ^ { 5 } b ^ { 4 } d ^ { 3 } ) ^ { 2 } : ( - \frac { 8 } { 15 } a ^ { 2 } b ^ { 2 } d ) ^ { 2 } : ( - \frac { 15 } { 2 } a ^ { 2 } b d ) ^ { 2 }$$

Evaluate

$$\frac{\left(abd\right)^{2}}{36}$$

Solution Steps

Expand \left(-\frac{2}{3}a^{5}b^{4}d^{3}\right)^{2}.

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}\left(a^{5}\right)^{2}\left(b^{4}\right)^{2}\left(d^{3}\right)^{2}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}a^{10}\left(b^{4}\right)^{2}\left(d^{3}\right)^{2}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}a^{10}b^{8}\left(d^{3}\right)^{2}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Calculate -\frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Expand \left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}.

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}\right)^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}\right)^{2}a^{4}\left(b^{2}\right)^{2}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}\right)^{2}a^{4}b^{4}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Calculate -\frac{8}{15} to the power of 2 and get \frac{64}{225}.

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\frac{64}{225}a^{4}b^{4}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\frac{4}{9}b^{4}d^{4}a^{6}}{\frac{64}{225}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Divide \frac{4}{9}b^{4}d^{4}a^{6} by \frac{64}{225} by multiplying \frac{4}{9}b^{4}d^{4}a^{6} by the reciprocal of \frac{64}{225}.

$$\frac{\frac{\frac{4}{9}b^{4}d^{4}a^{6}\times 225}{64}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Multiply \frac{4}{9} and 225 to get 100.

$$\frac{\frac{100b^{4}d^{4}a^{6}}{64}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Divide 100b^{4}d^{4}a^{6} by 64 to get \frac{25}{16}b^{4}d^{4}a^{6}.

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Expand \left(-\frac{15}{2}a^{2}bd\right)^{2}.

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\left(-\frac{15}{2}\right)^{2}\left(a^{2}\right)^{2}b^{2}d^{2}}$$

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

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\left(-\frac{15}{2}\right)^{2}a^{4}b^{2}d^{2}}$$

Calculate -\frac{15}{2} to the power of 2 and get \frac{225}{4}.

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\frac{225}{4}a^{4}b^{2}d^{2}}$$

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

$$\frac{\frac{25}{16}a^{2}b^{2}d^{2}}{\frac{225}{4}}$$

Divide \frac{25}{16}a^{2}b^{2}d^{2} by \frac{225}{4} by multiplying \frac{25}{16}a^{2}b^{2}d^{2} by the reciprocal of \frac{225}{4}.

$$\frac{\frac{25}{16}a^{2}b^{2}d^{2}\times 4}{225}$$

Multiply \frac{25}{16} and 4 to get \frac{25}{4}.

$$\frac{\frac{25}{4}a^{2}b^{2}d^{2}}{225}$$

Divide \frac{25}{4}a^{2}b^{2}d^{2} by 225 to get \frac{1}{36}a^{2}b^{2}d^{2}.

$$\frac{1}{36}a^{2}b^{2}d^{2}$$

Expand

$$\frac{\left(abd\right)^{2}}{36}$$

Solution Steps

Expand \left(-\frac{2}{3}a^{5}b^{4}d^{3}\right)^{2}.

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}\left(a^{5}\right)^{2}\left(b^{4}\right)^{2}\left(d^{3}\right)^{2}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}a^{10}\left(b^{4}\right)^{2}\left(d^{3}\right)^{2}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}a^{10}b^{8}\left(d^{3}\right)^{2}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\left(-\frac{2}{3}\right)^{2}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Calculate -\frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Expand \left(-\frac{8}{15}a^{2}b^{2}d\right)^{2}.

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}\right)^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}\right)^{2}a^{4}\left(b^{2}\right)^{2}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\left(-\frac{8}{15}\right)^{2}a^{4}b^{4}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Calculate -\frac{8}{15} to the power of 2 and get \frac{64}{225}.

$$\frac{\frac{\frac{4}{9}a^{10}b^{8}d^{6}}{\frac{64}{225}a^{4}b^{4}d^{2}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

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

$$\frac{\frac{\frac{4}{9}b^{4}d^{4}a^{6}}{\frac{64}{225}}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Divide \frac{4}{9}b^{4}d^{4}a^{6} by \frac{64}{225} by multiplying \frac{4}{9}b^{4}d^{4}a^{6} by the reciprocal of \frac{64}{225}.

$$\frac{\frac{\frac{4}{9}b^{4}d^{4}a^{6}\times 225}{64}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Multiply \frac{4}{9} and 225 to get 100.

$$\frac{\frac{100b^{4}d^{4}a^{6}}{64}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Divide 100b^{4}d^{4}a^{6} by 64 to get \frac{25}{16}b^{4}d^{4}a^{6}.

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\left(-\frac{15}{2}a^{2}bd\right)^{2}}$$

Expand \left(-\frac{15}{2}a^{2}bd\right)^{2}.

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\left(-\frac{15}{2}\right)^{2}\left(a^{2}\right)^{2}b^{2}d^{2}}$$

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

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\left(-\frac{15}{2}\right)^{2}a^{4}b^{2}d^{2}}$$

Calculate -\frac{15}{2} to the power of 2 and get \frac{225}{4}.

$$\frac{\frac{25}{16}b^{4}d^{4}a^{6}}{\frac{225}{4}a^{4}b^{2}d^{2}}$$

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

$$\frac{\frac{25}{16}a^{2}b^{2}d^{2}}{\frac{225}{4}}$$

Divide \frac{25}{16}a^{2}b^{2}d^{2} by \frac{225}{4} by multiplying \frac{25}{16}a^{2}b^{2}d^{2} by the reciprocal of \frac{225}{4}.

$$\frac{\frac{25}{16}a^{2}b^{2}d^{2}\times 4}{225}$$

Multiply \frac{25}{16} and 4 to get \frac{25}{4}.

$$\frac{\frac{25}{4}a^{2}b^{2}d^{2}}{225}$$

Divide \frac{25}{4}a^{2}b^{2}d^{2} by 225 to get \frac{1}{36}a^{2}b^{2}d^{2}.

$$\frac{1}{36}a^{2}b^{2}d^{2}$$