https://file.aiquickdraw.com/mathbot/file/1729699741531vye3jxcd.jpg

Let's break down and solve the expression from the image:

\[
5^3 + 4^3 \cdot 4^7 - \frac{12^{10}}{3^{10}}
\]

### Step-by-step solution:

1. **Simplify each part:**
   - \(5^3 = 5 \times 5 \times 5 = 125\)
   - \(4^3 = 4 \times 4 \times 4 = 64\)
   - \(4^7 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 16384\)
   - The fraction \(\frac{12^{10}}{3^{10}}\) can be rewritten as:
     \[
     \frac{12^{10}}{3^{10}} = \left(\frac{12}{3}\right)^{10} = 4^{10}
     \]
     Now, calculate \(4^{10} = 1048576\).

2. **Now, substitute the simplified terms into the expression:**
   \[
   5^3 + 4^3 \cdot 4^7 - \frac{12^{10}}{3^{10}} = 125 + 64 \cdot 16384 - 1048576
   \]

3. **Perform multiplication and subtraction:**
   - First, calculate \(64 \cdot 16384 = 1048576\)
   - So, the expression becomes:
     \[
     125 + 1048576 - 1048576
     \]
   - This simplifies to:
     \[
     125 + 0 = 125
     \]

### Final Answer:
\[
\boxed{125}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would this expression change if you used different base values for the powers?
2. What if the fraction was reversed, i.e., \(\frac{3^{10}}{12^{10}}\)?
3. Can you simplify \(4^{10}\) without using a calculator, through factorization?
4. How would the solution differ if the exponent on \(12\) and \(3\) was not equal, say \(12^8\) over \(3^{10}\)?
5. How could you verify the result using logarithmic properties?

**Tip:** When simplifying powers and exponents in fractions, try to find common bases to simplify the expression more easily!

Solve the expression 5^3 + 4^3 * 4^7 - (12^10 / 3^10)

Solve a complex exponential expression involving powers and fraction simplification. Learn how to break down and simplify terms like 5^3, 4^3 * 4^7, and (12^10 / 3^10) step by step, suitable for Grades 10-12 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