Solve frac{(2x^{4}y^{2}*2^{2}x^{3}y)^{2}(2^{4}x^{2}y^{4}div2^{2}xy)^{3}}{(2^5x^4y^2)^2}

$$\frac { ( 2 x ^ { 4 } y ^ { 2 } \times 2 ^ { 2 } x ^ { 3 } y ) ^ { 2 } ( 2 ^ { 4 } x ^ { 2 } y ^ { 4 } \div 2 ^ { 2 } x y ) ^ { 3 } } { ( 2 ^ { 5 } x ^ { 4 } y ^ { 2 } ) ^ { 2 } }$$

Evaluate

$$4x^{15}y^{17}$$

Solution Steps

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

$$\frac{\left(2^{3}x^{4}y^{2}x^{3}y\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{\left(2^{3}x^{7}y^{2}y\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{\left(2^{3}x^{7}y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\left(8x^{7}y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Expand \left(8x^{7}y^{3}\right)^{2}.

$$\frac{8^{2}\left(x^{7}\right)^{2}\left(y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{8^{2}x^{14}\left(y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{8^{2}x^{14}y^{6}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{64x^{14}y^{6}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{64x^{14}y^{6}\times \left(2^{2}x^{2}y^{4}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{64x^{14}y^{6}\times \left(2^{2}x^{3}y^{4}y\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{64x^{14}y^{6}\times \left(2^{2}x^{3}y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Expand \left(2^{2}x^{3}y^{5}\right)^{3}.

$$\frac{64x^{14}y^{6}\times \left(2^{2}\right)^{3}\left(x^{3}\right)^{3}\left(y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{64x^{14}y^{6}\times 2^{6}\left(x^{3}\right)^{3}\left(y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{64x^{14}y^{6}\times 2^{6}x^{9}\left(y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{64x^{14}y^{6}\times 2^{6}x^{9}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{64x^{14}y^{6}\times 64x^{9}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Multiply 64 and 64 to get 4096.

$$\frac{4096x^{14}y^{6}x^{9}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 14 and 9 to get 23.

$$\frac{4096x^{23}y^{6}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 6 and 15 to get 21.

$$\frac{4096x^{23}y^{21}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{4096x^{23}y^{21}}{\left(32x^{4}y^{2}\right)^{2}}$$

Expand \left(32x^{4}y^{2}\right)^{2}.

$$\frac{4096x^{23}y^{21}}{32^{2}\left(x^{4}\right)^{2}\left(y^{2}\right)^{2}}$$

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

$$\frac{4096x^{23}y^{21}}{32^{2}x^{8}\left(y^{2}\right)^{2}}$$

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

$$\frac{4096x^{23}y^{21}}{32^{2}x^{8}y^{4}}$$

Calculate 32 to the power of 2 and get 1024.

$$\frac{4096x^{23}y^{21}}{1024x^{8}y^{4}}$$

Cancel out 1024y^{4}x^{8} in both numerator and denominator.

$$4x^{15}y^{17}$$

Expand

$$4x^{15}y^{17}$$

Solution Steps

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

$$\frac{\left(2^{3}x^{4}y^{2}x^{3}y\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{\left(2^{3}x^{7}y^{2}y\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{\left(2^{3}x^{7}y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\left(8x^{7}y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Expand \left(8x^{7}y^{3}\right)^{2}.

$$\frac{8^{2}\left(x^{7}\right)^{2}\left(y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{8^{2}x^{14}\left(y^{3}\right)^{2}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{8^{2}x^{14}y^{6}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{64x^{14}y^{6}\times \left(\frac{2^{4}x^{2}y^{4}}{2^{2}}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{64x^{14}y^{6}\times \left(2^{2}x^{2}y^{4}xy\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{64x^{14}y^{6}\times \left(2^{2}x^{3}y^{4}y\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{64x^{14}y^{6}\times \left(2^{2}x^{3}y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Expand \left(2^{2}x^{3}y^{5}\right)^{3}.

$$\frac{64x^{14}y^{6}\times \left(2^{2}\right)^{3}\left(x^{3}\right)^{3}\left(y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

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

$$\frac{64x^{14}y^{6}\times 2^{6}\left(x^{3}\right)^{3}\left(y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{64x^{14}y^{6}\times 2^{6}x^{9}\left(y^{5}\right)^{3}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{64x^{14}y^{6}\times 2^{6}x^{9}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{64x^{14}y^{6}\times 64x^{9}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Multiply 64 and 64 to get 4096.

$$\frac{4096x^{14}y^{6}x^{9}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 14 and 9 to get 23.

$$\frac{4096x^{23}y^{6}y^{15}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 6 and 15 to get 21.

$$\frac{4096x^{23}y^{21}}{\left(2^{5}x^{4}y^{2}\right)^{2}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{4096x^{23}y^{21}}{\left(32x^{4}y^{2}\right)^{2}}$$

Expand \left(32x^{4}y^{2}\right)^{2}.

$$\frac{4096x^{23}y^{21}}{32^{2}\left(x^{4}\right)^{2}\left(y^{2}\right)^{2}}$$

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

$$\frac{4096x^{23}y^{21}}{32^{2}x^{8}\left(y^{2}\right)^{2}}$$

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

$$\frac{4096x^{23}y^{21}}{32^{2}x^{8}y^{4}}$$

Calculate 32 to the power of 2 and get 1024.

$$\frac{4096x^{23}y^{21}}{1024x^{8}y^{4}}$$

Cancel out 1024y^{4}x^{8} in both numerator and denominator.

$$4x^{15}y^{17}$$