Solve [(xy/2)^{2}]^{3}+3/2(x^{2}y)^2*(1/2xy^2-1/4xy^2)^2

$$[ ( \frac { x y } { 2 } ) ^ { 2 } ] ^ { 3 } + \frac { 3 } { 2 } ( x ^ { 2 } y ) ^ { 2 } \cdot ( \frac { 1 } { 2 } x y ^ { 2 } - \frac { 1 } { 4 } x y ^ { 2 } ) ^ { 2 }$$

Evaluate

$$\frac{7\left(xy\right)^{6}}{64}$$

Solution Steps

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

$$\left(\frac{xy}{2}\right)^{6}+\frac{3}{2}\left(x^{2}y\right)^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

To raise \frac{xy}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}\left(x^{2}y\right)^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

Expand \left(x^{2}y\right)^{2}.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}\left(x^{2}\right)^{2}y^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

Combine \frac{1}{2}xy^{2} and -\frac{1}{4}xy^{2} to get \frac{1}{4}xy^{2}.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \left(\frac{1}{4}xy^{2}\right)^{2}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \left(\frac{1}{4}\right)^{2}x^{2}\left(y^{2}\right)^{2}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \left(\frac{1}{4}\right)^{2}x^{2}y^{4}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \frac{1}{16}x^{2}y^{4}$$

Multiply \frac{3}{2} and \frac{1}{16} to get \frac{3}{32}.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{32}x^{4}y^{2}x^{2}y^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{32}x^{6}y^{2}y^{4}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{32}x^{6}y^{6}$$

Expand \left(xy\right)^{6}.

$$\frac{x^{6}y^{6}}{2^{6}}+\frac{3}{32}x^{6}y^{6}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{x^{6}y^{6}}{64}+\frac{3}{32}x^{6}y^{6}$$

Combine \frac{x^{6}y^{6}}{64} and \frac{3}{32}x^{6}y^{6} to get \frac{7}{64}x^{6}y^{6}.

$$\frac{7}{64}x^{6}y^{6}$$

Expand

$$\frac{7\left(xy\right)^{6}}{64}$$

Solution Steps

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

$$\left(\frac{xy}{2}\right)^{6}+\frac{3}{2}\left(x^{2}y\right)^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

To raise \frac{xy}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}\left(x^{2}y\right)^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

Expand \left(x^{2}y\right)^{2}.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}\left(x^{2}\right)^{2}y^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\left(\frac{1}{2}xy^{2}-\frac{1}{4}xy^{2}\right)^{2}$$

Combine \frac{1}{2}xy^{2} and -\frac{1}{4}xy^{2} to get \frac{1}{4}xy^{2}.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \left(\frac{1}{4}xy^{2}\right)^{2}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \left(\frac{1}{4}\right)^{2}x^{2}\left(y^{2}\right)^{2}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \left(\frac{1}{4}\right)^{2}x^{2}y^{4}$$

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

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{2}x^{4}y^{2}\times \frac{1}{16}x^{2}y^{4}$$

Multiply \frac{3}{2} and \frac{1}{16} to get \frac{3}{32}.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{32}x^{4}y^{2}x^{2}y^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{32}x^{6}y^{2}y^{4}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\left(xy\right)^{6}}{2^{6}}+\frac{3}{32}x^{6}y^{6}$$

Expand \left(xy\right)^{6}.

$$\frac{x^{6}y^{6}}{2^{6}}+\frac{3}{32}x^{6}y^{6}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{x^{6}y^{6}}{64}+\frac{3}{32}x^{6}y^{6}$$

Combine \frac{x^{6}y^{6}}{64} and \frac{3}{32}x^{6}y^{6} to get \frac{7}{64}x^{6}y^{6}.

$$\frac{7}{64}x^{6}y^{6}$$