Answer :

GinnyAnswer
Certainly! Let's solve the given equation step-by-step.

The given equation is:

[tex]\[ \frac{1}{3^{5x - 2}} = 81^{x + 4} \][/tex]

### Step 1: Express 81 as a power of 3

We know that [tex]\(81\)[/tex] can be written as [tex]\(3^4\)[/tex]. Therefore,

[tex]\[ 81 = 3^4 \][/tex]

Thus, the equation becomes:

[tex]\[ \frac{1}{3^{5x - 2}} = (3^4)^{x + 4} \][/tex]

### Step 2: Simplify the right-hand side

According to the laws of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Therefore,

[tex]\[ (3^4)^{x + 4} = 3^{4(x + 4)} \][/tex]

Simplify further:

[tex]\[ 3^{4(x + 4)} = 3^{4x + 16} \][/tex]

So now the equation is:

[tex]\[ \frac{1}{3^{5x - 2}} = 3^{4x + 16} \][/tex]

### Step 3: Rewrite the left-hand side

The left-hand side can be rewritten using the property of exponents [tex]\( \frac{1}{a^m} = a^{-m} \)[/tex]:

[tex]\[ \frac{1}{3^{5x - 2}} = 3^{-(5x - 2)} \][/tex]

### Step 4: Set the exponents equal

Since the bases are the same (both are [tex]\(3\)[/tex]), we can equate the exponents:

[tex]\[ -(5x - 2) = 4x + 16 \][/tex]

Simplify the equation:

[tex]\[ -5x + 2 = 4x + 16 \][/tex]

Combine like terms:

[tex]\[ -5x - 4x = 16 - 2 \][/tex]

[tex]\[ -9x = 14 \][/tex]

### Step 5: Solve for [tex]\(x\)[/tex]

Divide both sides by [tex]\(-9\)[/tex]:

[tex]\[ x = \frac{14}{-9} \][/tex]

Simplify the fraction:

[tex]\[ x = -\frac{14}{9} \][/tex]

Or as a decimal:

[tex]\[ x = -1.5555555555555556 \][/tex]

This is the solution to the equation.
Hi1315

Answer:

[tex]x = -\dfrac{14}{9}[/tex]

Step-by-step explanation:

To solve [tex]\dfrac{1}{3^{5x-2}} = 81^{x+4}[/tex]

1. Rewrite 81 as [tex]3^4:[/tex]

[tex]81^{x+4} = (3^4)^{x+4} = 3^{4(x+4)}[/tex]

2. Simplify:

[tex]3^{4(x+4)} = 3^{4x + 16}[/tex]

3. Compare exponents:

[tex]\dfrac{1}{3^{5x-2}} = 3^{-(5x-2)}\\\\ 3^{-(5x-2)} = 3^{4x + 16}\\\\ -(5x - 2) = 4x + 16\\\\ -5x + 2 = 4x + 16\\\\ -14 = 9x[/tex]

[tex]x = -\dfrac{14}{9}[/tex]

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