To solve the expression [tex]\(12^{-\frac{4}{7}}\)[/tex] and match it with the table values step-by-step, let's break this down carefully.
### Step-by-Step Solution
1. Understanding the Expression:
- The given expression is [tex]\(12^{-\frac{4}{7}}\)[/tex].
- A negative exponent indicates the reciprocal of the positive exponent value. So, [tex]\(12^{-\frac{4}{7}} = \frac{1}{12^{\frac{4}{7}}}\)[/tex].
2. Finding the Positive Exponent Value:
- Compute [tex]\(12^{\frac{4}{7}}\)[/tex].
- This value is used to simplify and identify the corresponding term in the table. Numerically, this equals approximately [tex]\(4.136899759811512\)[/tex].
3. Reciprocal of the Positive Exponent Value:
- Since [tex]\(12^{-\frac{4}{7}} = \frac{1}{12^{\frac{4}{7}}}\)[/tex], we take the reciprocal.
- Numerically, [tex]\(\frac{1}{12^{\frac{4}{7}}} = \frac{1}{4.136899759811512} \approx 0.2417269109864926\)[/tex].
4. Match with the Table:
- Examining the table, we look for the row-wise and column-wise components that match our computations.
- We observe:
- [tex]\(\sqrt[7]{12^4}\)[/tex] in the first row represents [tex]\(12^{\frac{4}{7}}\)[/tex].
- [tex]\(\frac{1}{\sqrt[7]{12^4}}\)[/tex] in the third row represents [tex]\(\frac{1}{12^{\frac{4}{7}}}\)[/tex].
- The third row, first column, matches our computed reciprocal value of [tex]\(12^{\frac{4}{7}}\)[/tex], which is approximately [tex]\(0.2417269109864926\)[/tex].
### Conclusion:
Based on the detailed breakdown:
- The equivalent match for [tex]\(12^{-\frac{4}{7}}\)[/tex] in the provided table is found in the third row, first column: [tex]\(\frac{1}{\sqrt[7]{12^4}}\)[/tex].
- Numerically, this matches the value [tex]\( \approx 0.2417269109864926 \)[/tex].
Thus, the value of [tex]\(12^{-\frac{4}{7}}\)[/tex] is approximately [tex]\(0.2417269109864926\)[/tex].
