Solve (2/3a^{2}-b^{2})(2/3a^2+b^2)(4/9a^4+b^4)

$$( \frac { 2 } { 3 } a ^ { 2 } - b ^ { 2 } ) ( \frac { 2 } { 3 } a ^ { 2 } + b ^ { 2 } ) ( \frac { 4 } { 9 } a ^ { 4 } + b ^ { 4 } )$$

Evaluate

$$\frac{16a^{8}}{81}-b^{8}$$

Solution Steps

Use the distributive property to multiply \frac{2}{3}a^{2}-b^{2} by \frac{2}{3}a^{2}+b^{2} and combine like terms.

$$\left(\frac{4}{9}a^{4}-b^{4}\right)\left(\frac{4}{9}a^{4}+b^{4}\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{4}{9}a^{4}\right)^{2}-\left(b^{4}\right)^{2}$$

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

$$\left(\frac{4}{9}a^{4}\right)^{2}-b^{8}$$

Expand \left(\frac{4}{9}a^{4}\right)^{2}.

$$\left(\frac{4}{9}\right)^{2}\left(a^{4}\right)^{2}-b^{8}$$

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

$$\left(\frac{4}{9}\right)^{2}a^{8}-b^{8}$$

Calculate \frac{4}{9} to the power of 2 and get \frac{16}{81}.

$$\frac{16}{81}a^{8}-b^{8}$$

Expand

$$\frac{16a^{8}}{81}-b^{8}$$

Solution Steps

Use the distributive property to multiply \frac{2}{3}a^{2}-b^{2} by \frac{2}{3}a^{2}+b^{2} and combine like terms.

$$\left(\frac{4}{9}a^{4}-b^{4}\right)\left(\frac{4}{9}a^{4}+b^{4}\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{4}{9}a^{4}\right)^{2}-\left(b^{4}\right)^{2}$$

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

$$\left(\frac{4}{9}a^{4}\right)^{2}-b^{8}$$

Expand \left(\frac{4}{9}a^{4}\right)^{2}.

$$\left(\frac{4}{9}\right)^{2}\left(a^{4}\right)^{2}-b^{8}$$

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

$$\left(\frac{4}{9}\right)^{2}a^{8}-b^{8}$$

Calculate \frac{4}{9} to the power of 2 and get \frac{16}{81}.

$$\frac{16}{81}a^{8}-b^{8}$$