Let's solve each expression step-by-step to determine the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
### Determining the value of [tex]\(x\)[/tex]:
We start with the expression:
[tex]\[(-j)^{-7} = \frac{1}{(-j)^2}\][/tex]
1. Simplify the left side:
[tex]\[(-j)^{-7} = \frac{1}{(-j)^7}\][/tex]
Since [tex]\((-j)^7 = (-1)^7 \cdot j^7 = -j^7\)[/tex], it follows that:
[tex]\[\frac{1}{(-j)^7} = \frac{1}{-j^7} = -\frac{1}{j^7}\][/tex]
2. Simplify the right side:
[tex]\[\frac{1}{(-j)^2} = \frac{1}{-j^2}\][/tex]
Since [tex]\((-j)^2 = (-1)^2 \cdot j^2 = j^2\)[/tex], it follows that:
[tex]\[\frac{1}{(-j)^2} = \frac{1}{j^2}\][/tex]
Now, setting the simplified left side equal to the simplified right side:
[tex]\[-\frac{1}{j^7} = \frac{1}{j^2}\][/tex]
We can cross-multiply to clear the fractions:
[tex]\[-1 \cdot j^2 = 1 \cdot j^7\][/tex]
[tex]\[ -j^2 = j^7\][/tex]
To solve for [tex]\(j\)[/tex], divide both sides by [tex]\(j^2\)[/tex]:
[tex]\[-1 = j^5\][/tex]
[tex]\[j^5 = -1\][/tex]
Thus:
[tex]\[ x = j^5 \][/tex]
### Determining the value of [tex]\(y\)[/tex]:
We start with the expression:
[tex]\[k^{-5} + m^{-10} = \frac{1}{k^3} + \frac{1}{m^{10}}\][/tex]
1. Let's rewrite [tex]\(k^{-5}\)[/tex] and [tex]\(m^{-10}\)[/tex]:
[tex]\[k^{-5} = \frac{1}{k^5}\][/tex]
[tex]\[m^{-10} = \frac{1}{m^{10}}\][/tex]
2. Now look at each part of the right-hand side expression:
[tex]\[\frac{1}{k^3} + \frac{1}{m^{10}}\][/tex]
Clearly, the term on the right involving [tex]\(m^{-10}\)[/tex] directly matches:
[tex]\[ m^{-10} = \frac{1}{m^{10}} \][/tex]
For the right-hand side, we need to find a match for [tex]\(k^{-5}\)[/tex] with [tex]\(\frac{1}{k^3}\)[/tex]. However, [tex]\(\frac{1}{k^3}\)[/tex] does not directly match [tex]\(\frac{1}{k^5}\)[/tex]:
To reconcile this, let's consider and make the right-hand side simpler:
[tex]\[ k^{-5} = k^{-5} \][/tex]
[tex]\[ \frac{1}{k^3} = k^{-3}\][/tex]
By seeing both parts, it seems the left simplified need to meet equality:
Therefore, since [tex]\(k^{-5}\)[/tex] and [tex]\(\frac{1}{k^3}\)[/tex] don't give a direct result:
And [tex]\(m^{-10}\)[/tex] already consistent meeting all parts, thus
[tex]\[ k^{-5}=0 \][/tex]
Thus:
[tex]\[ y = 0 \][/tex]
So, the values for the variables in the simplified expressions are:
[tex]\[ x = j^5 \][/tex]
[tex]\[ y = 0 \][/tex]
