Answer :

To solve the equation

[tex]\[ \left(\frac{-2}{5}\right)^{-3} \times \left(\frac{-2}{5}\right)^{11} = \left(\frac{-2}{5}\right)^{3x + 2}, \][/tex]

we can use the properties of exponents. Specifically, when multiplying two expressions with the same base, we add their exponents. Thus, the left-hand side of the equation can be simplified as follows:

[tex]\[ \left(\frac{-2}{5}\right)^{-3} \times \left(\frac{-2}{5}\right)^{11} = \left(\frac{-2}{5}\right)^{-3 + 11}. \][/tex]

Next, we calculate the exponent on the left-hand side:

[tex]\[ -3 + 11 = 8. \][/tex]

So, the equation now becomes:

[tex]\[ \left(\frac{-2}{5}\right)^{8} = \left(\frac{-2}{5}\right)^{3x + 2}. \][/tex]

Since the bases are the same on both sides of the equation, we can set the exponents equal to each other:

[tex]\[ 8 = 3x + 2. \][/tex]

Now, solve for [tex]\(x\)[/tex]. First, subtract 2 from both sides of the equation:

[tex]\[ 8 - 2 = 3x, \][/tex]

which simplifies to:

[tex]\[ 6 = 3x. \][/tex]

Next, divide both sides by 3:

[tex]\[ x = \frac{6}{3}. \][/tex]

Thus, we find:

[tex]\[ x = 2. \][/tex]

Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(x = 2\)[/tex].