Let's closely analyze Tamika's work to identify where she went wrong.
Given the expression:
[tex]\[
\frac{18 a ^{-5} b^{-6}}{30 a ^3 b^{-6}}
\][/tex]
Tamika's work steps:
1. She simplified the coefficients:
[tex]\[
\frac{18}{30} = \frac{3}{5}
\][/tex]
This step is correct.
2. She worked with the exponents for [tex]\( a \)[/tex]:
[tex]\[
a^{-5} \div a^3
\][/tex]
According to the rules of exponents, when dividing with the same base, we subtract the exponents:
[tex]\[
a^{-5} \div a^3 = a^{-5 - 3} = a^{-8}
\][/tex]
However, in Tamika's solution, she obtained:
[tex]\[
a^{-2}
\][/tex]
This suggests that she did not correctly subtract the exponents for [tex]\( a \)[/tex].
3. She worked with the exponents for [tex]\( b \)[/tex]:
[tex]\[
b^{-6} \div b^{-6}
\][/tex]
Again, according to the rules of exponents, subtracting same bases:
[tex]\[
b^{-6 - (-6)} = b^0 = 1
\][/tex]
This step is correct as [tex]\( b^{-6} \div b^{-6} \)[/tex] cancels out to 1.
Now, re-examine Tamika's intermediate result:
[tex]\[
\frac{3 a^{-2}}{5 b^{11}}
\][/tex]
To reach:
[tex]\[
\frac{3}{5 a^2 b^{11}}
\][/tex]
We see that Tamika moved the base with a negative exponent correctly but made an error with initially obtaining [tex]\( a^{-2} \)[/tex] instead of [tex]\( a^{-8} \)[/tex].
Let's outline the correct steps again:
1. Simplify the coefficients:
[tex]\[
\frac{18}{30} = \frac{3}{5}
\][/tex]
2. Subtract the exponents for [tex]\( a \)[/tex]:
[tex]\[
a^{-5} \div a^3 = a^{-8}
\][/tex]
3. Subtract the exponents for [tex]\( b \)[/tex]:
[tex]\[
b^{-6} \div b^{-6} = b^0 = 1
\][/tex]
Putting it all together:
[tex]\[
\frac{3 a^{-8}}{5} = \frac{3}{5 a^8}
\][/tex]
Thus, the correct simplified form is:
[tex]\[
\frac{3}{5 a^8}
\][/tex]
Therefore, Tamika's error was:
She divided the exponents.
