Sure! Let's solve for [tex]\( x \)[/tex] in the given equation:
[tex]\[ (-19)^{10} \times (-19)^4 = (-19)^{3x - 1} \][/tex]
### Step 1: Simplify the left-hand side
We start by applying the properties of exponents to simplify the left-hand side of the equation. One key property is that when you multiply two exponents with the same base, you add the exponents:
[tex]\[ a^m \times a^n = a^{m+n} \][/tex]
Using this property, we can simplify the left-hand side:
[tex]\[ (-19)^{10} \times (-19)^4 = (-19)^{10 + 4} \][/tex]
This gives us:
[tex]\[ (-19)^{14} \][/tex]
### Step 2: Set up the simplified equation
Now we rewrite the equation with the simplified left-hand side:
[tex]\[ (-19)^{14} = (-19)^{3x - 1} \][/tex]
### Step 3: Equate the exponents
Since the bases are the same on both sides of the equation, we can set their exponents equal to each other:
[tex]\[ 14 = 3x - 1 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
To find [tex]\( x \)[/tex], we need to isolate it on one side of the equation. Start by adding 1 to both sides to get rid of the -1 on the right side:
[tex]\[ 14 + 1 = 3x \][/tex]
So:
[tex]\[ 15 = 3x \][/tex]
Next, divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{15}{3} \][/tex]
This simplifies to:
[tex]\[ x = 5 \][/tex]
### Conclusion
The value of [tex]\( x \)[/tex] is:
[tex]\[ x = 5 \][/tex]
