11. $$ 100^{5 x+2}=\left(\frac{1}{10}\right)^{11-x} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
100^{5x + 2} = \left(\dfrac{1}{10}\right)^{11 - x}
$$

**Step 1: Express Both Sides with the Same Base**

First, recognize that:
- $$100 = 10^2$$
- $$\dfrac{1}{10} = 10^{-1}$$

Rewrite both sides using base 10:

$$
100^{5x + 2} = (10^2)^{5x + 2} = 10^{2(5x + 2)} = 10^{10x + 4}
$$

$$
\left(\dfrac{1}{10}\right)^{11 - x} = (10^{-1})^{11 - x} = 10^{- (11 - x)} = 10^{x - 11}
$$

Now, the equation becomes:

$$
10^{10x + 4} = 10^{x - 11}
$$

**Step 2: Set the Exponents Equal**

Since the bases are the same and $$10 > 1$$, the exponents must be equal:

$$
10x + 4 = x - 11
$$

**Step 3: Solve for $$x$$**

Subtract $$x$$ from both sides:

$$
10x - x + 4 = -11 \
9x + 4 = -11
$$

Subtract 4 from both sides:

$$
9x = -11 - 4 \
9x = -15
$$

Divide both sides by 9:

$$
x = \dfrac{-15}{9} = -\dfrac{5}{3}
$$

**Solution:**
$$
x = -\dfrac{5}{3}
$$
$$ x = -\dfrac{5}{3} $$

11. \( 100^{5 x+2}=\left(\frac{1}{10}\right)^{11-x} \)