Simplify :
`((-1/2)^8)/((-1/2)^5)-((-1/2)^5)/((-1/2)^3)`

Answer

To simplify the expression \(\frac{(-\frac{1}{2})^8}{(-\frac{1}{2})^5} - \frac{(-\frac{1}{2})^5}{(-\frac{1}{2})^3}\), we can follow these steps:

### Step 1: Apply the property of exponents
Using the property \(\frac{a^n}{a^m} = a^{n-m}\), we can simplify each term.

1. For the first term:
\[ \frac{(-\frac{1}{2})^8}{(-\frac{1}{2})^5} = (-\frac{1}{2})^{8-5} = (-\frac{1}{2})^3 \]

2. For the second term:
\[ \frac{(-\frac{1}{2})^5}{(-\frac{1}{2})^3} = (-\frac{1}{2})^{5-3} = (-\frac{1}{2})^2 \]

So, the expression now becomes:
\[ (-\frac{1}{2})^3 - (-\frac{1}{2})^2 \]

### Step 2: Calculate the powers
Now we calculate the powers:
1. \((- \frac{1}{2})^3 = -\frac{1}{8}\)
2. \((- \frac{1}{2})^2 = \frac{1}{4}\)

So, we can rewrite the expression as:
\[ -\frac{1}{8} - \frac{1}{4} \]

### Step 3: Find a common denominator
To subtract these fractions, we need a common denominator. The least common multiple of 8 and 4 is 8.

1. Rewrite \(\frac{1}{4}\) with a denominator of 8:
\[ \frac{1}{4} = \frac{2}{8} \]

### Step 4: Perform the subtraction
Now we can perform the subtraction:
\[ -\frac{1}{8} - \frac{2}{8} = -\frac{1 + 2}{8} = -\frac{3}{8} \]

### Final Answer
Thus, the simplified expression is:
\[ -\frac{3}{8} \]

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