Solve (2c^{3}d^{2})^{5}(frac{c^{8}d^8}{4c^2d})^3quad2^5

$$( 2 c ^ { 3 } d ^ { 2 } ) ^ { 5 } ( \frac { c ^ { 8 } d ^ { 8 } } { 4 c ^ { 2 } d } ) ^ { 3 } \quad 2 ^ { 5 }$$

Evaluate

$$16d^{31}c^{33}$$

Solution Steps

Expand \left(2c^{3}d^{2}\right)^{5}.

$$2^{5}\left(c^{3}\right)^{5}\left(d^{2}\right)^{5}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

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

$$2^{5}c^{15}\left(d^{2}\right)^{5}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

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

$$2^{5}c^{15}d^{10}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

Calculate 2 to the power of 5 and get 32.

$$32c^{15}d^{10}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

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

$$32c^{15}d^{10}\times \left(\frac{c^{6}d^{7}}{4}\right)^{3}\times 2^{5}$$

To raise \frac{c^{6}d^{7}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$32c^{15}d^{10}\times \frac{\left(c^{6}d^{7}\right)^{3}}{4^{3}}\times 2^{5}$$

Calculate 2 to the power of 5 and get 32.

$$32c^{15}d^{10}\times \frac{\left(c^{6}d^{7}\right)^{3}}{4^{3}}\times 32$$

Multiply 32 and 32 to get 1024.

$$1024c^{15}d^{10}\times \frac{\left(c^{6}d^{7}\right)^{3}}{4^{3}}$$

Expand \left(c^{6}d^{7}\right)^{3}.

$$1024c^{15}d^{10}\times \frac{\left(c^{6}\right)^{3}\left(d^{7}\right)^{3}}{4^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$1024c^{15}d^{10}\times \frac{c^{18}\left(d^{7}\right)^{3}}{4^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$1024c^{15}d^{10}\times \frac{c^{18}d^{21}}{4^{3}}$$

Calculate 4 to the power of 3 and get 64.

$$1024c^{15}d^{10}\times \frac{c^{18}d^{21}}{64}$$

Cancel out 64, the greatest common factor in 1024 and 64.

$$16c^{18}d^{21}c^{15}d^{10}$$

To multiply powers of the same base, add their exponents. Add 18 and 15 to get 33.

$$16c^{33}d^{21}d^{10}$$

To multiply powers of the same base, add their exponents. Add 21 and 10 to get 31.

$$16c^{33}d^{31}$$

Expand

$$16d^{31}c^{33}$$

Solution Steps

Expand \left(2c^{3}d^{2}\right)^{5}.

$$2^{5}\left(c^{3}\right)^{5}\left(d^{2}\right)^{5}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

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

$$2^{5}c^{15}\left(d^{2}\right)^{5}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

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

$$2^{5}c^{15}d^{10}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

Calculate 2 to the power of 5 and get 32.

$$32c^{15}d^{10}\times \left(\frac{c^{8}d^{8}}{4c^{2}d}\right)^{3}\times 2^{5}$$

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

$$32c^{15}d^{10}\times \left(\frac{c^{6}d^{7}}{4}\right)^{3}\times 2^{5}$$

To raise \frac{c^{6}d^{7}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$32c^{15}d^{10}\times \frac{\left(c^{6}d^{7}\right)^{3}}{4^{3}}\times 2^{5}$$

Calculate 2 to the power of 5 and get 32.

$$32c^{15}d^{10}\times \frac{\left(c^{6}d^{7}\right)^{3}}{4^{3}}\times 32$$

Multiply 32 and 32 to get 1024.

$$1024c^{15}d^{10}\times \frac{\left(c^{6}d^{7}\right)^{3}}{4^{3}}$$

Expand \left(c^{6}d^{7}\right)^{3}.

$$1024c^{15}d^{10}\times \frac{\left(c^{6}\right)^{3}\left(d^{7}\right)^{3}}{4^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$1024c^{15}d^{10}\times \frac{c^{18}\left(d^{7}\right)^{3}}{4^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$1024c^{15}d^{10}\times \frac{c^{18}d^{21}}{4^{3}}$$

Calculate 4 to the power of 3 and get 64.

$$1024c^{15}d^{10}\times \frac{c^{18}d^{21}}{64}$$

Cancel out 64, the greatest common factor in 1024 and 64.

$$16c^{18}d^{21}c^{15}d^{10}$$

To multiply powers of the same base, add their exponents. Add 18 and 15 to get 33.

$$16c^{33}d^{21}d^{10}$$

To multiply powers of the same base, add their exponents. Add 21 and 10 to get 31.

$$16c^{33}d^{31}$$