Solve (frac{a^{2}}{3b})^3*(c/ab)^2*(b^3/ac)^2=

$$( \frac { a ^ { 2 } } { 3 b } ) ^ { 3 } \cdot ( \frac { c } { a b } ) ^ { 2 } \cdot ( \frac { b ^ { 3 } } { a c } ) ^ { 2 } =$$

Evaluate

$$\frac{ba^{2}}{27}$$

Solution Steps

To raise \frac{a^{2}}{3b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}}\times \left(\frac{c}{ab}\right)^{2}\times \left(\frac{b^{3}}{ac}\right)^{2}$$

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

$$\frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}}\times \frac{c^{2}}{\left(ab\right)^{2}}\times \left(\frac{b^{3}}{ac}\right)^{2}$$

To raise \frac{b^{3}}{ac} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}}\times \frac{c^{2}}{\left(ab\right)^{2}}\times \frac{\left(b^{3}\right)^{2}}{\left(ac\right)^{2}}$$

Multiply \frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}} times \frac{c^{2}}{\left(ab\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}\right)^{3}c^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}}\times \frac{\left(b^{3}\right)^{2}}{\left(ac\right)^{2}}$$

Multiply \frac{\left(a^{2}\right)^{3}c^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}} times \frac{\left(b^{3}\right)^{2}}{\left(ac\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}\right)^{3}c^{2}\left(b^{3}\right)^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

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

$$\frac{a^{6}c^{2}\left(b^{3}\right)^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

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

$$\frac{a^{6}c^{2}b^{6}}{\left(3b\right)^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

Expand \left(3b\right)^{3}.

$$\frac{a^{6}c^{2}b^{6}}{3^{3}b^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{a^{6}c^{2}b^{6}}{27b^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

Expand \left(ab\right)^{2}.

$$\frac{a^{6}c^{2}b^{6}}{27b^{3}a^{2}b^{2}\left(ac\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{a^{6}c^{2}b^{6}}{27b^{5}a^{2}\left(ac\right)^{2}}$$

Expand \left(ac\right)^{2}.

$$\frac{a^{6}c^{2}b^{6}}{27b^{5}a^{2}a^{2}c^{2}}$$

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

$$\frac{a^{6}c^{2}b^{6}}{27b^{5}a^{4}c^{2}}$$

Cancel out c^{2}a^{4}b^{5} in both numerator and denominator.

$$\frac{ba^{2}}{27}$$

Expand

$$\frac{ba^{2}}{27}$$

Solution Steps

To raise \frac{a^{2}}{3b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}}\times \left(\frac{c}{ab}\right)^{2}\times \left(\frac{b^{3}}{ac}\right)^{2}$$

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

$$\frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}}\times \frac{c^{2}}{\left(ab\right)^{2}}\times \left(\frac{b^{3}}{ac}\right)^{2}$$

To raise \frac{b^{3}}{ac} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}}\times \frac{c^{2}}{\left(ab\right)^{2}}\times \frac{\left(b^{3}\right)^{2}}{\left(ac\right)^{2}}$$

Multiply \frac{\left(a^{2}\right)^{3}}{\left(3b\right)^{3}} times \frac{c^{2}}{\left(ab\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}\right)^{3}c^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}}\times \frac{\left(b^{3}\right)^{2}}{\left(ac\right)^{2}}$$

Multiply \frac{\left(a^{2}\right)^{3}c^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}} times \frac{\left(b^{3}\right)^{2}}{\left(ac\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}\right)^{3}c^{2}\left(b^{3}\right)^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

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

$$\frac{a^{6}c^{2}\left(b^{3}\right)^{2}}{\left(3b\right)^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

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

$$\frac{a^{6}c^{2}b^{6}}{\left(3b\right)^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

Expand \left(3b\right)^{3}.

$$\frac{a^{6}c^{2}b^{6}}{3^{3}b^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{a^{6}c^{2}b^{6}}{27b^{3}\left(ab\right)^{2}\left(ac\right)^{2}}$$

Expand \left(ab\right)^{2}.

$$\frac{a^{6}c^{2}b^{6}}{27b^{3}a^{2}b^{2}\left(ac\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{a^{6}c^{2}b^{6}}{27b^{5}a^{2}\left(ac\right)^{2}}$$

Expand \left(ac\right)^{2}.

$$\frac{a^{6}c^{2}b^{6}}{27b^{5}a^{2}a^{2}c^{2}}$$

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

$$\frac{a^{6}c^{2}b^{6}}{27b^{5}a^{4}c^{2}}$$

Cancel out c^{2}a^{4}b^{5} in both numerator and denominator.

$$\frac{ba^{2}}{27}$$