Examine Tamika's work below. What is Tamika's error?

[tex]\[
\begin{aligned}
\frac{18 a ^{-5} b^{-6}}{30 a ^3 b^{-6}} & = \frac{3 a ^{-2} b^{-11}}{5} \\
& = \frac{3}{5 a ^2 b^{11}}
\end{aligned}
\][/tex]

A. She added the exponents.
B. She simplified the coefficients incorrectly.
C. She divided the exponents.
D. She moved bases with positive exponents.



Answer :

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.