21) $$ \frac{4^{2 x-1}}{4^{2 x}}=4^{3 x} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{4^{2x-1}}{4^{2x}} = 4^{3x}
$$

**Step 1: Simplify the Left Side**

Using the properties of exponents, specifically $$\frac{a^m}{a^n} = a^{m-n}$$, we simplify the left side:

$$
\frac{4^{2x-1}}{4^{2x}} = 4^{(2x-1) - 2x} = 4^{-1} = \frac{1}{4}
$$

So the equation becomes:

$$
\frac{1}{4} = 4^{3x}
$$

**Step 2: Express $$\frac{1}{4}$$ with Base 4**

Notice that $$\frac{1}{4}$$ can be written as $$4^{-1}$$:

$$
4^{-1} = 4^{3x}
$$

**Step 3: Equate the Exponents**

Since the bases are the same and the equation $$4^a = 4^b$$ implies $$a = b$$, we set the exponents equal to each other:

$$
-1 = 3x
$$

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

$$
x = \frac{-1}{3}
$$

**Final Answer:**

$$
x = -\frac{1}{3}
$$
$$ x = -\frac{1}{3} $$

21) \( \frac{4^{2 x-1}}{4^{2 x}}=4^{3 x} \)