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Solve for [tex]\( x \)[/tex].

[tex]\[ 6^{2x+2} \cdot 6^{3x} = 1 \][/tex]

[tex]\[ x = \][/tex]

Sagot :

GinnyAnswer
Let's solve the equation [tex]\(6^{2x + 2} \cdot 6^{3x} = 1\)[/tex] step-by-step.

1. Combine Exponents:
Given the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can combine the exponents on the left-hand side.

[tex]\[ 6^{(2x + 2)} \cdot 6^{3x} = 6^{(2x + 2 + 3x)} = 6^{(5x + 2)} \][/tex]

So the equation now looks like:

[tex]\[ 6^{5x + 2} = 1 \][/tex]

2. Interpret the Exponential Equation:
Recall that any non-zero number raised to the power of 0 is 1. Hence, we need the exponent on the left to be zero for the entire term to equal 1.

[tex]\[ 6^{0} = 1 \][/tex]

Therefore, we set the exponent equal to 0:

[tex]\[ 5x + 2 = 0 \][/tex]

3. Solve for [tex]\(x\)[/tex]:

Subtract 2 from both sides:

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

Divide both sides by 5:

[tex]\[ x = \frac{-2}{5} \][/tex]

Hence,

[tex]\[ x = -0.4 \][/tex]

So the solution to the equation [tex]\(6^{2x + 2} \cdot 6^{3x} = 1\)[/tex] is
[tex]\[ x = -0.4 \][/tex]
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