If `2= 10^(m) and 3= 10^(n)`, then find the value of `0.15`.

Options

A. `10^(n- m +1)`
B. `10^(n+ m+ 1)`
C. `10^(n- m - 1)`
D. `10^(- (n-m-1) )`

Answer

To solve the problem step by step, we start with the given equations and the value we want to find.

### Step 1: Understand the given equations
We have:
- \( 2 = 10^m \)
- \( 3 = 10^n \)

We need to find the value of \( 0.15 \).

### Step 2: Express \( 0.15 \) in a simpler form
We can express \( 0.15 \) as a fraction:
\[ 0.15 = \frac{15}{100} \]
To simplify this, we can rewrite \( 100 \) as \( 10^2 \):
\[ 0.15 = \frac{15}{10^2} \]

### Step 3: Simplify \( 15 \)
Next, we can express \( 15 \) as \( \frac{3}{2} \times 10 \):
\[ 0.15 = \frac{3 \times 10}{2 \times 10^2} = \frac{3}{2} \times \frac{1}{10} \]

### Step 4: Substitute the values of \( 2 \) and \( 3 \)
Now we can substitute \( 3 \) and \( 2 \) with their respective powers of \( 10 \):
\[ 0.15 = \frac{10^n}{10^m} \times \frac{1}{10} \]

### Step 5: Combine the powers of \( 10 \)
Using the property of exponents, we can combine the powers:
\[ 0.15 = \frac{10^n}{10^m \times 10^1} = \frac{10^n}{10^{m+1}} \]

### Step 6: Apply the property of division of powers
Now, we can apply the property of division of powers:
\[ 0.15 = 10^{n - (m + 1)} = 10^{n - m - 1} \]

### Step 7: Conclusion
Thus, we find that:
\[ 0.15 = 10^{n - m - 1} \]