Solve frac{45(20x^{20}y^{18}z^{32})^{2}}{(3x^7y^10z^23)^5}

$$\frac { 45 ( 20 x ^ { 20 } y ^ { 18 } z ^ { 32 } ) ^ { 2 } } { ( 3 x ^ { 7 } y ^ { 10 } z ^ { 23 } ) ^ { 5 } }$$

Evaluate

$$\frac{2000x^{5}}{27y^{14}z^{51}}$$

Solution Steps

Expand \left(20x^{20}y^{18}z^{32}\right)^{2}.

$$\frac{45\times 20^{2}\left(x^{20}\right)^{2}\left(y^{18}\right)^{2}\left(z^{32}\right)^{2}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 20 and 2 to get 40.

$$\frac{45\times 20^{2}x^{40}\left(y^{18}\right)^{2}\left(z^{32}\right)^{2}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 18 and 2 to get 36.

$$\frac{45\times 20^{2}x^{40}y^{36}\left(z^{32}\right)^{2}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 32 and 2 to get 64.

$$\frac{45\times 20^{2}x^{40}y^{36}z^{64}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{45\times 400x^{40}y^{36}z^{64}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

Multiply 45 and 400 to get 18000.

$$\frac{18000x^{40}y^{36}z^{64}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

Expand \left(3x^{7}y^{10}z^{23}\right)^{5}.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}\left(x^{7}\right)^{5}\left(y^{10}\right)^{5}\left(z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 5 to get 35.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}x^{35}\left(y^{10}\right)^{5}\left(z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and 5 to get 50.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}x^{35}y^{50}\left(z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 23 and 5 to get 115.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}x^{35}y^{50}z^{115}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{18000x^{40}y^{36}z^{64}}{243x^{35}y^{50}z^{115}}$$

Cancel out 9x^{35}y^{36}z^{64} in both numerator and denominator.

$$\frac{2000x^{5}}{27y^{14}z^{51}}$$

Expand

$$\frac{2000x^{5}}{27y^{14}z^{51}}$$

Solution Steps

Expand \left(20x^{20}y^{18}z^{32}\right)^{2}.

$$\frac{45\times 20^{2}\left(x^{20}\right)^{2}\left(y^{18}\right)^{2}\left(z^{32}\right)^{2}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 20 and 2 to get 40.

$$\frac{45\times 20^{2}x^{40}\left(y^{18}\right)^{2}\left(z^{32}\right)^{2}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 18 and 2 to get 36.

$$\frac{45\times 20^{2}x^{40}y^{36}\left(z^{32}\right)^{2}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 32 and 2 to get 64.

$$\frac{45\times 20^{2}x^{40}y^{36}z^{64}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{45\times 400x^{40}y^{36}z^{64}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

Multiply 45 and 400 to get 18000.

$$\frac{18000x^{40}y^{36}z^{64}}{\left(3x^{7}y^{10}z^{23}\right)^{5}}$$

Expand \left(3x^{7}y^{10}z^{23}\right)^{5}.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}\left(x^{7}\right)^{5}\left(y^{10}\right)^{5}\left(z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 5 to get 35.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}x^{35}\left(y^{10}\right)^{5}\left(z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and 5 to get 50.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}x^{35}y^{50}\left(z^{23}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 23 and 5 to get 115.

$$\frac{18000x^{40}y^{36}z^{64}}{3^{5}x^{35}y^{50}z^{115}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{18000x^{40}y^{36}z^{64}}{243x^{35}y^{50}z^{115}}$$

Cancel out 9x^{35}y^{36}z^{64} in both numerator and denominator.

$$\frac{2000x^{5}}{27y^{14}z^{51}}$$