Let's go through the problem step-by-step and identify Anja's mistake.
We start with the given expression:
[tex]\[
\frac{10 x^{-5}}{-5 x^{10}}
\][/tex]
### Step 1: Simplify the Coefficients
The coefficients are 10 and -5. To simplify these, we divide them:
[tex]\[
\frac{10}{-5} = -2
\][/tex]
### Step 2: Simplify the Exponents
We have exponents of [tex]\(x\)[/tex] in the numerator and the denominator. When dividing exponents with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
x^{-5} \div x^{10} = x^{-5 - 10} = x^{-15}
\][/tex]
### Step 3: Combine the Simplified Coefficient and Exponent
So, the simplified expression becomes:
[tex]\[
\frac{10 x^{-5}}{-5 x^{10}} = -2 x^{-15}
\][/tex]
### Step 4: Rewriting in Standard Form
In standard form, this would be written as:
[tex]\[
-2 x^{-15} = -\frac{2}{x^{15}}
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
-\frac{2}{x^{15}}
\][/tex]
### Identify Anja's Mistake
Anja simplified the expression to:
[tex]\[
\frac{15}{x^{15}}
\][/tex]
Comparing Anja's result with the correct result, we can see her mistake. The correct numerator is [tex]\(-2\)[/tex], but Anja has it as 15.
It appears Anja treated the coefficients incorrectly. Instead of dividing the coefficients (as we did [tex]\(\frac{10}{-5} = -2\)[/tex]), she subtracted them (resulting in [tex]\(10 - (-5) = 10 + 5 = 15\)[/tex]). Thus, her mistake was:
[tex]\[
\text{She subtracted the coefficients instead of dividing them.}
\][/tex]
In conclusion, the correct answer is:
She subtracted the coefficients instead of dividing them.
