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Complete the equation by supplying the missing exponent.

[tex]\[ 3^3 \cdot 3^{-6} = 3^x \][/tex]

A. [tex]\( x = 2 \)[/tex]

B. [tex]\( x = -3 \)[/tex]

C. [tex]\( x = 4 \)[/tex]

D. [tex]\( x = -8 \)[/tex]

Sagot :

GinnyAnswer
Sure, let's solve the equation step-by-step.

The given equation is:

[tex]\[ 3^3 \cdot 3^{-6} = 3^2 \][/tex]

To solve this, notice that the left-hand side combines the exponents of the same base (3). According to the properties of exponents, when you multiply powers of the same base, you add the exponents together:

[tex]\[ 3^3 \cdot 3^{-6} = 3^{3 + (-6)} \][/tex]

So, the combined exponent on the left-hand side is:

[tex]\[ 3 + (-6) = -3 \][/tex]

Therefore, the equation becomes:

[tex]\[ 3^{-3} = 3^2 \][/tex]

For the equation to hold true, we need to match the exponents. To do that, we equate the combined exponent to the exponent on the right-hand side, which is 2. So:

[tex]\[ -3 + \text{(missing exponent)} = 2 \][/tex]

To find the missing exponent, we solve for it:

[tex]\[ \text{(missing exponent)} = 2 + 3 \][/tex]
[tex]\[ \text{(missing exponent)} = 5 \][/tex]

So the completed equation is:

[tex]\[ 3^3 \cdot 3^{-6} \cdot 3^5 = 3^2 \][/tex]

The missing exponent that completes the equation is:

5

Therefore, the answer is 5.
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