Answer :
Let's simplify the given expression step-by-step. We need to simplify the expression:
[tex]\[ 6 \cdot 6^{t-2} \cdot 6^t \][/tex]
We use the properties of exponents to do this.
### Step 1: Understand Exponents Multiplication Rule
The multiplication rule for exponents states that \(a^m \cdot a^n = a^{m+n}\). This means we can combine terms with the same base by adding their exponents.
### Step 2: Apply the Rule to Combine Exponents
Let's combine \(6^{t-2} \cdot 6^t\):
[tex]\[ 6^{t-2} \cdot 6^t = 6^{(t-2) + t} \][/tex]
### Step 3: Simplify the Exponents
Now, add the exponents in the exponent:
[tex]\[ (t-2) + t = t + t - 2 = 2t - 2 \][/tex]
So, we have:
[tex]\[ 6^{t-2} \cdot 6^t = 6^{2t - 2} \][/tex]
### Step 4: Combine with the Constant Term
Now we need to include the remaining \(6\) from the original expression. Notice that \(6 = 6^1\):
[tex]\[ 6 \cdot 6^{2t-2} = 6^1 \cdot 6^{2t-2} \][/tex]
Again, use the exponents multiplication rule \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 6^1 \cdot 6^{2t-2} = 6^{1 + (2t-2)} \][/tex]
### Step 5: Simplify the Final Exponent
Simplify the exponent:
[tex]\[ 1 + (2t - 2) = 1 + 2t - 2 = 2t - 1 \][/tex]
Therefore, the simplified expression is:
[tex]\[ 6^{2t - 1} \][/tex]
### Conclusion
Among the given choices:
- \(6^{2t - 1}\)
- \(18^{2t - 2}\)
- \(216^{2t - 1}\)
- \(6^{2t - 2}\)
The correct simplified expression is:
[tex]\[ 6^{2t - 1} \][/tex]
[tex]\[ 6 \cdot 6^{t-2} \cdot 6^t \][/tex]
We use the properties of exponents to do this.
### Step 1: Understand Exponents Multiplication Rule
The multiplication rule for exponents states that \(a^m \cdot a^n = a^{m+n}\). This means we can combine terms with the same base by adding their exponents.
### Step 2: Apply the Rule to Combine Exponents
Let's combine \(6^{t-2} \cdot 6^t\):
[tex]\[ 6^{t-2} \cdot 6^t = 6^{(t-2) + t} \][/tex]
### Step 3: Simplify the Exponents
Now, add the exponents in the exponent:
[tex]\[ (t-2) + t = t + t - 2 = 2t - 2 \][/tex]
So, we have:
[tex]\[ 6^{t-2} \cdot 6^t = 6^{2t - 2} \][/tex]
### Step 4: Combine with the Constant Term
Now we need to include the remaining \(6\) from the original expression. Notice that \(6 = 6^1\):
[tex]\[ 6 \cdot 6^{2t-2} = 6^1 \cdot 6^{2t-2} \][/tex]
Again, use the exponents multiplication rule \(a^m \cdot a^n = a^{m+n}\):
[tex]\[ 6^1 \cdot 6^{2t-2} = 6^{1 + (2t-2)} \][/tex]
### Step 5: Simplify the Final Exponent
Simplify the exponent:
[tex]\[ 1 + (2t - 2) = 1 + 2t - 2 = 2t - 1 \][/tex]
Therefore, the simplified expression is:
[tex]\[ 6^{2t - 1} \][/tex]
### Conclusion
Among the given choices:
- \(6^{2t - 1}\)
- \(18^{2t - 2}\)
- \(216^{2t - 1}\)
- \(6^{2t - 2}\)
The correct simplified expression is:
[tex]\[ 6^{2t - 1} \][/tex]
