6. $$ 25\left(125^{-x}\right)=625 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
25 \cdot \left(125^{-x}\right) = 625
$$

**Step 1: Express All Terms with the Same Base**

Notice that 25, 125, and 625 are all powers of 5:

- $$25 = 5^2$$
- $$125 = 5^3$$
- $$625 = 5^4$$

Substitute these into the equation:

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

**Step 2: Simplify the Equation**

Use the property of exponents $$(a^m)^n = a^{m \cdot n}$$:

$$
5^2 \cdot 5^{-3x} = 5^4
$$

Combine the exponents on the left side using $$a^m \cdot a^n = a^{m+n}$$:

$$
5^{2 - 3x} = 5^4
$$

**Step 3: Set Exponents Equal**

Since the bases are the same, set the exponents equal to each other:

$$
2 - 3x = 4
$$

**Step 4: Solve for $$x$$**

$$
-3x = 4 - 2
$$
$$
-3x = 2
$$
$$
x = \frac{2}{-3} = -\frac{2}{3}
$$

**Conclusion**

The solution to the equation is:

$$
x = -\frac{2}{3}
$$
$$ x = -\frac{2}{3} $$

6. \( 25\left(125^{-x}\right)=625 \)