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Applying the rules of exponents and radicals, the table can be completed as follows:
First column: [tex]64^1 = 64[/tex]
Second Column: [tex]\mathbf{64^\frac{1}{2} = 8}[/tex]
Third Column: [tex]\mathbf{64^\frac{1}{3} = 4}[/tex]
Fourth Column: [tex]\mathbf{64^0 = 1}[/tex]
Fifth Column: [tex]\mathbf{64^{-\frac{1}{2}} = \frac{1}{8}}[/tex]
Sixth Column: [tex]\mathbf{64^{-1} = \frac{1}{64}}[/tex]
The table to complete is attached below.
We will need to apply the knowledge of exponents and radicals of a number in order to fill out the table.
The numbers expressed as powers of 64 in the top row are equivalent to the numbers expressed as radicals or rational numbers in the corresponding cells in the bottom row.
Also, recall the following rules of exponents and radicals:
Appling the above rule, we would have the following:
[tex]64^1 = 64[/tex]
[tex]64^\frac{1}{2} = \sqrt[]{64} = 8\\\\\mathbf{64^\frac{1}{2} = 8}[/tex]
[tex]64^\frac{1}{3} = \sqrt[3]{64} = 4\\\\\mathbf{64^\frac{1}{3} = 4}[/tex]
[tex]\mathbf{64^0 = 1}[/tex]
[tex]64^{-\frac{1}{2}} = \frac{1}{64^{\frac{1}{2}} } = \frac{1}{\sqrt{64} } = \frac{1}{8} \\\\\\\mathbf{64^{-\frac{1}{2}} = \frac{1}{8}}[/tex]
[tex]64^{-1} = \frac{1}{64^1} } = \frac{1}{64 } \\\\\mathbf{64^{-1} = \frac{1}{64}}[/tex]
Therefore, applying the rules of exponents and radicals, the table can be completed as follows:
First column: [tex]64^1 = 64[/tex]
Second Column: [tex]\mathbf{64^\frac{1}{2} = 8}[/tex]
Third Column: [tex]\mathbf{64^\frac{1}{3} = 4}[/tex]
Fourth Column: [tex]\mathbf{64^0 = 1}[/tex]
Fifth Column: [tex]\mathbf{64^{-\frac{1}{2}} = \frac{1}{8}}[/tex]
Sixth Column: [tex]\mathbf{64^{-1} = \frac{1}{64}}[/tex]
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