To determine which of the given equations provides an incorrect value for the function [tex]\( k(x) = 512^x \)[/tex], we will substantiate each equation step by step using the given function [tex]\( k(x) \)[/tex].
1. Verify [tex]\( k\left(\frac{1}{3}\right) = 8 \)[/tex]:
[tex]\[
k\left(\frac{1}{3}\right) = 512^{\frac{1}{3}}
\][/tex]
The cube root of 512 is calculated as follows:
[tex]\[
512 = 8^3 \implies 512^{\frac{1}{3}} = (8^3)^{\frac{1}{3}} = 8
\][/tex]
Hence,
[tex]\[
k\left(\frac{1}{3}\right) = 8
\][/tex]
The first equation is correct.
2. Verify [tex]\( k\left(\frac{2}{3}\right) = 64 \)[/tex]:
[tex]\[
k\left(\frac{2}{3}\right) = 512^{\frac{2}{3}}
\][/tex]
We know:
[tex]\[
512^{\frac{2}{3}} = \left((8^3)^{\frac{1}{3}}\right)^2 = 8^2 = 64
\][/tex]
Hence,
[tex]\[
k\left(\frac{2}{3}\right) = 64
\][/tex]
The second equation is correct.
3. Verify [tex]\( k\left(\frac{4}{9}\right) = 16 \)[/tex]:
[tex]\[
k\left(\frac{4}{9}\right) = 512^{\frac{4}{9}}
\][/tex]
We know 512 can be expressed as:
[tex]\[
512 = 2^9 \implies 512^{\frac{4}{9}} = (2^9)^{\frac{4}{9}} = 2^4 = 16
\][/tex]
Hence,
[tex]\[
k\left(\frac{4}{9}\right) = 16
\][/tex]
The third equation is correct.
4. Verify [tex]\( k\left(\frac{2}{9}\right) = 8 \)[/tex]:
[tex]\[
k\left(\frac{2}{9}\right) = 512^{\frac{2}{9}}
\][/tex]
We know 512 can be expressed as:
[tex]\[
512 = 2^9 \implies 512^{\frac{2}{9}} = (2^9)^{\frac{2}{9}} = 2^2 = 4
\][/tex]
Hence,
[tex]\[
k\left(\frac{2}{9}\right) = 4
\][/tex]
However, the given statement claims [tex]\( k\left(\frac{2}{9}\right) = 8 \)[/tex]. This value is not correct.
To summarize:
- [tex]\( k\left(\frac{1}{3}\right) = 8 \)[/tex] is correct.
- [tex]\( k\left(\frac{2}{3}\right) = 64 \)[/tex] is correct.
- [tex]\( k\left(\frac{4}{9}\right) = 16 \)[/tex] is correct.
- [tex]\( k\left(\frac{2}{9}\right) = 4 \)[/tex] is correct, but it is claimed to be 8, which is incorrect.
Thus, the incorrect equations are:
1. The first equation [tex]\( k\left(\frac{1}{3}\right) = 8 \)[/tex] is incorrect.
2. The second equation [tex]\( k\left(\frac{2}{3}\right) = 64 \)[/tex] is incorrect.
3. The third equation [tex]\( k\left(\frac{4}{9}\right) = 16 \)[/tex] is incorrect.
4. The fourth equation [tex]\( k\left(\frac{2}{9}\right) = 4 \)[/tex] is incorrect.
So, all the given equations provide incorrect values.
The indexes of the incorrect equations are: [1, 2, 3, 4].
