Use like bases to solve the exponential equation. $$ \left(\frac{1}{64}\right)^{3 n} \cdot 8=2^{12} $$

Question

UpStudy AI · Accepted Answer

To solve the exponential equation:

$$
\left(\frac{1}{64}\right)^{3n} \cdot 8 = 2^{12}
$$

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

First, recognize that all numbers can be expressed as powers of 2:

- $$ \frac{1}{64} = 2^{-6} $$ because $$ 2^6 = 64 $$.
- $$ 8 = 2^3 $$.

So, the equation becomes:

$$
\left(2^{-6}\right)^{3n} \cdot 2^3 = 2^{12}
$$

**Step 2: Simplify the Left Side**

Apply the power of a power rule $$(a^m)^n = a^{m \cdot n}$$:

$$
2^{-6 \cdot 3n} \cdot 2^3 = 2^{-18n} \cdot 2^3
$$

Now, use the multiplication rule for exponents $$a^m \cdot a^n = a^{m+n}$$:

$$
2^{-18n + 3} = 2^{12}
$$

**Step 3: Set the Exponents Equal**

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

$$
-18n + 3 = 12
$$

**Step 4: Solve for $$n$$**

Subtract 3 from both sides:

$$
-18n = 12 - 3
$$
$$
-18n = 9
$$

Divide both sides by $$-18$$:

$$
n = \frac{9}{-18} = -\frac{1}{2}
$$

**Final Answer:**

$$
n = -\frac{1}{2}
$$
$$ n = -\frac{1}{2} $$

Use like bases to solve the exponential equation.