Solve [(a^{2}-x^2)^-3]^-2*[(a+x)(a-x)]^-5

$$[ ( a ^ { 2 } - x ^ { 2 } ) ^ { - 3 } ] ^ { - 2 } \cdot [ ( a + x ) ( a - x ) ] ^ { - 5 }$$

Evaluate

$$a^{2}-x^{2}$$

Solution Steps

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

$$\left(a^{2}-x^{2}\right)^{6}\left(\left(a+x\right)\left(a-x\right)\right)^{-5}$$

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

$$\left(a^{2}-x^{2}\right)^{6}\left(a^{2}-x^{2}\right)^{-5}$$

To multiply powers of the same base, add their exponents. Add 6 and -5 to get 1.

$$\left(a^{2}-x^{2}\right)^{1}$$

Calculate a^{2}-x^{2} to the power of 1 and get a^{2}-x^{2}.

$$a^{2}-x^{2}$$

Expand

$$a^{2}-x^{2}$$

Solution Steps

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

$$\left(a^{2}-x^{2}\right)^{6}\left(\left(a+x\right)\left(a-x\right)\right)^{-5}$$

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

$$\left(a^{2}-x^{2}\right)^{6}\left(a^{2}-x^{2}\right)^{-5}$$

To multiply powers of the same base, add their exponents. Add 6 and -5 to get 1.

$$\left(a^{2}-x^{2}\right)^{1}$$

Calculate a^{2}-x^{2} to the power of 1 and get a^{2}-x^{2}.

$$a^{2}-x^{2}$$