To find the value of [tex]\( m \)[/tex] in the equation
[tex]\[
\left(\frac{3}{5}\right)^{2 m} \times \left(\frac{9}{25}\right)^3 = \left(\frac{3}{5}\right)^{-2},
\][/tex]
we will follow a series of steps to simplify and solve the equation.
### Step 1: Rewrite [tex]\( \left( \frac{9}{25} \right)^3 \)[/tex]
First, observe that [tex]\( \frac{9}{25} \)[/tex] can be expressed as [tex]\( \left( \frac{3}{5} \right)^2 \)[/tex]:
[tex]\[
\left( \frac{9}{25} \right)^3 = \left( \left( \frac{3}{5} \right)^2 \right)^3.
\][/tex]
### Step 2: Simplify the exponentiation
We know that [tex]\( \left( a^b \right)^c = a^{bc} \)[/tex]:
[tex]\[
\left( \left( \frac{3}{5} \right)^2 \right)^3 = \left( \frac{3}{5} \right)^{2 \times 3} = \left( \frac{3}{5} \right)^6.
\][/tex]
### Step 3: Substitute and combine exponents
Now substitute [tex]\( \left( \frac{3}{5} \right)^6 \)[/tex] back into the original equation:
[tex]\[
\left( \frac{3}{5} \right)^{2m} \times \left( \frac{3}{5} \right)^6 = \left( \frac{3}{5} \right)^{-2}.
\][/tex]
Next, use the property of exponents that [tex]\( a^b \times a^c = a^{b+c} \)[/tex]:
[tex]\[
\left( \frac{3}{5} \right)^{2m + 6} = \left( \frac{3}{5} \right)^{-2}.
\][/tex]
### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2m + 6 = -2.
\][/tex]
### Step 5: Solve for [tex]\( m \)[/tex]
Solve the equation for [tex]\( m \)[/tex]:
[tex]\[
2m + 6 = -2.
\][/tex]
Subtract 6 from both sides:
[tex]\[
2m = -2 - 6,
\][/tex]
which simplifies to:
[tex]\[
2m = -8.
\][/tex]
Divide by 2:
[tex]\[
m = -4.
\][/tex]
### Conclusion
The value of [tex]\( m \)[/tex] is:
[tex]\[
\boxed{-4}.
\][/tex]
