Convert between exponentials with powers and a product of exponents Precalculus

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

2^{3\cdot x}=\left( 2^{3} \right)^{x}

Using the definition of a power:

2^{3}=2\cdot2\cdot2=8

We conclude that:

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

3^{4\cdot x}=\left( 3^{4} \right)^{x}

Using the definition of a power:

3^{4}=3\cdot3\cdot3\cdot3=81

We conclude that:

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

2^{-3\cdot x}=\left( 2^{-3} \right)^{x}

Using the definition of a power:

2^{-3}=\dfrac{1}{2\cdot2\cdot2}=\dfrac{1}{8}

We conclude that:

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

\left( \dfrac{1}{5} \right)^{-2\cdot x}=\left( \left( \dfrac{1}{5} \right)^{-2} \right)^{x}

Using the definition of a power:

\left( \dfrac{1}{5} \right)^{-2}=5\cdot5=25

We conclude that:

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

10^{2\cdot x}=\left( 10^{2} \right)^{x}

Using the definition of a power:

10^{2}=10\cdot10=100

We conclude that:

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

4^{-3\cdot x}=\left( 4^{-3} \right)^{x}

Using the definition of a power:

4^{-3}=\dfrac{1}{4\cdot4\cdot4}=\dfrac{1}{64}

We conclude that:

For any two integers m and n :

\left( x^{m} \right)^{n}=x^{m\cdot n}

In our problem, applying the formula from right to left:

\left( \dfrac{2}{3} \right)^{-x}=\left( \left( \dfrac{2}{3} \right)^{-1} \right)^{x}

Using the definition of a power:

\left( \dfrac{2}{3} \right)^{-1}=\dfrac{3}{2}

We conclude that: