Solve T=frac{x^{4}*(x^{2})^{3}*x^{-3}*x^6}{(x^5*x^-2)^3}

$$T = \frac { x ^ { 4 } \cdot ( x ^ { 2 } ) ^ { 3 } \cdot x ^ { - 3 } \cdot x ^ { 6 } } { ( x ^ { 5 } \cdot x ^ { - 2 } ) ^ { 3 } }$$

Solve for T

$$T=x^{4}$$

$$x\neq 0$$

Solution Steps

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

$$T=\frac{x^{4}x^{6}x^{-3}x^{6}}{\left(x^{5}x^{-2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 6 to get 10.

$$T=\frac{x^{10}x^{-3}x^{6}}{\left(x^{5}x^{-2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 10 and -3 to get 7.

$$T=\frac{x^{7}x^{6}}{\left(x^{5}x^{-2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 7 and 6 to get 13.

$$T=\frac{x^{13}}{\left(x^{5}x^{-2}\right)^{3}}$$

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

$$T=\frac{x^{13}}{\left(x^{3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$T=\frac{x^{13}}{x^{9}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 9 from 13 to get 4.

$$T=x^{4}$$

Solve for x (complex solution)

$$x=i\sqrt[4]{T}$$

$$x=\sqrt[4]{T}$$

$$x=-\sqrt[4]{T}$$

$$x=-i\sqrt[4]{T}\text{, }T\neq 0$$

Solve for x

$$x=\sqrt[4]{T}$$

$$x=-\sqrt[4]{T}\text{, }T>0$$