https://file.aiquickdraw.com/mathbot/file/17267219509535w7scdgb.jpg

To simplify the given expression, we need to handle the fraction:

\[
\frac{\frac{\frac{1}{10^{105}}}{\frac{1}{10^{105}} + \frac{1.45 \times 10^{15}}{10^{60}}}}{\frac{\frac{1.45 \times 10^{15}}{10^{60}}}{\frac{1}{10^{105}} + \frac{1.45 \times 10^{15}}{10^{60}}}}
\]

First, let's rewrite and simplify each term:

### Step 1: Simplify the numerator

\[
\frac{1}{10^{105}} + \frac{1.45 \times 10^{15}}{10^{60}} = \frac{1}{10^{105}} + \frac{1.45 \times 10^{15 - 60}}{1} = \frac{1}{10^{105}} + 1.45 \times 10^{-45}
\]

### Step 2: Simplify the denominator

Since the denominator is exactly the same as the numerator, we have:

\[
\frac{1}{10^{105}} + \frac{1.45 \times 10^{15}}{10^{60}} = \frac{1}{10^{105}} + 1.45 \times 10^{-45}
\]

### Step 3: Final expression

The final expression is now:

\[
\frac{\frac{1}{\frac{1}{10^{105}} + 1.45 \times 10^{-45}}}{\frac{1.45 \times 10^{15}}{\frac{1}{10^{105}} + 1.45 \times 10^{-45}}}
\]

This can be simplified to:

\[
\frac{1}{1.45 \times 10^{15}} = \frac{1}{1.45 \times 10^{15}}
\]

Thus, the simplified form of the given complex fraction is:

\[
\frac{1}{1.45 \times 10^{15}}
\]

Would you like further details or have any questions on the steps?

**Relative Questions:**
1. How would you evaluate a similar expression with different exponents?
2. What is the impact of adding very small numbers like \(10^{-105}\) to larger numbers like \(1.45 \times 10^{-45}\)?
3. How does the order of operations affect the simplification process in such cases?
4. Can you simplify the expression if the base of the exponents were not 10?
5. How would this expression change if the term \(1.45 \times 10^{15}\) had a different coefficient?

**Tip:** Always ensure to align the exponents when dealing with such fractional expressions for easier simplification.

Simplify this complex fraction involving large exponents and coefficients: \( \frac{ \frac{1}{10^{105}} + \frac{1.45 \times 10^{15}}{10^{60}}}{ \frac{\frac{1.45 \times 10^{15}}{10^{60}}}{ \frac{1}{10^{105}} + \frac{1.45 \times 10^{15}}{10^{60}}}} \)

This problem demonstrates the simplification of a complex fraction involving large exponents and coefficients. The solution uses algebraic techniques, rules for adding numbers with exponents, and simplification of complex fractions, making it suitable for high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation