Certainly! Let's solve the equation step-by-step: [tex]\(\frac{1^2}{2} - \frac{1^{-1}}{4} = 2^{-3}\)[/tex].
### Step 1: Calculate [tex]\(\frac{1^2}{2}\)[/tex]
1. First, evaluate the exponent: [tex]\(1^2\)[/tex].
[tex]\[
1^2 = 1
\][/tex]
2. Next, divide by 2:
[tex]\[
\frac{1}{2} = 0.5
\][/tex]
So, [tex]\(\frac{1^2}{2} = 0.5\)[/tex].
### Step 2: Calculate [tex]\(\frac{1^{-1}}{4}\)[/tex]
1. Evaluate the exponent: [tex]\(1^{-1}\)[/tex].
[tex]\[
1^{-1} = \frac{1}{1} = 1
\][/tex]
2. Next, divide by 4:
[tex]\[
\frac{1}{4} = 0.25
\][/tex]
So, [tex]\(\frac{1^{-1}}{4} = 0.25\)[/tex].
### Step 3: Calculate [tex]\(2^{-3}\)[/tex]
1. Evaluate the exponent: [tex]\(2^{-3}\)[/tex].
[tex]\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125
\][/tex]
So, [tex]\(2^{-3} = 0.125\)[/tex].
### Step 4: Assemble the Left-Hand Side of the Equation
1. We already have [tex]\(\frac{1^2}{2} = 0.5\)[/tex].
2. And [tex]\(\frac{1^{-1}}{4} = 0.25\)[/tex].
3. Subtract [tex]\(\frac{1^{-1}}{4}\)[/tex] from [tex]\(\frac{1^2}{2}\)[/tex]:
[tex]\[
0.5 - 0.25 = 0.25
\][/tex]
### Step 5: Compare Both Sides
1. The left-hand side (LHS) of the equation is:
[tex]\[
LHS = 0.25
\][/tex]
2. The right-hand side (RHS) of the equation is:
[tex]\[
RHS = 0.125
\][/tex]
Given these results:
- Term 1: [tex]\(\frac{1^2}{2} = 0.5\)[/tex]
- Term 2: [tex]\(\frac{1^{-1}}{4} = 0.25\)[/tex]
- Term 3: [tex]\(2^{-3} = 0.125\)[/tex]
- LHS: [tex]\(0.25\)[/tex]
- RHS: [tex]\(0.125\)[/tex]
It's evident from the calculations, the LHS of the original equation is [tex]\(0.25\)[/tex], and the RHS is [tex]\(0.125\)[/tex]. Thus, [tex]\(0.25 \neq 0.125\)[/tex].
So, the equation [tex]\(\frac{1^2}{2} - \frac{1^{-1}}{4} \neq 2^{-3}\)[/tex].
