Math Problem Statement

1800 : { 450 : [450 - ( 4 . 5 mũ3 - 2 mũ3 . 5 mũ2)]}

Solution

Let's break down the expression step by step:

Expression:

1800:{450:[450(4.532352)]}1800 : \left\{ 450 : \left[ 450 - \left( 4.5^3 - 2^3 \cdot 5^2 \right) \right] \right\}

Step 1: Calculate the value inside the square brackets

Start with the expression 4.5323524.5^3 - 2^3 \cdot 5^2:

  • 4.53=4.5×4.5×4.5=91.1254.5^3 = 4.5 \times 4.5 \times 4.5 = 91.125
  • 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8
  • 52=5×5=255^2 = 5 \times 5 = 25
  • So, 2352=8×25=2002^3 \cdot 5^2 = 8 \times 25 = 200

Now, calculate the difference: 4.53(2352)=91.125200=108.8754.5^3 - (2^3 \cdot 5^2) = 91.125 - 200 = -108.875

Step 2: Substitute this value into the expression

Now the expression becomes: 1800:{450:[450(108.875)]}1800 : \left\{ 450 : [450 - (-108.875)] \right\}

Step 3: Simplify inside the parentheses

450(108.875)=450+108.875=558.875450 - (-108.875) = 450 + 108.875 = 558.875

Now the expression is: 1800:{450:558.875}1800 : \left\{ 450 : 558.875 \right\}

Step 4: Simplify the division

First, calculate 450:558.875450 : 558.875: 450÷558.8750.805450 \div 558.875 \approx 0.805

Step 5: Final division

Now, calculate 1800:0.8051800 : 0.805: 1800÷0.8052237.51800 \div 0.805 \approx 2237.5

Final Answer:

2237.5\boxed{2237.5}

Let me know if you'd like any further details on the steps!


Further Questions for Exploration:

  1. What is the difference between exponentiation and multiplication in algebraic expressions?
  2. How do order of operations affect the solution to a complex expression like this?
  3. What is the significance of negative numbers in real-world calculations?
  4. Can we apply logarithms to simplify expressions involving large exponents?
  5. How would rounding affect this calculation in practical scenarios?

Tip: Always handle the operations in parentheses first, as they often have the most significant impact on the final result!

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Order of Operations
Division
Algebraic Expressions

Formulas

Exponentiation: a^b
Order of operations: PEMDAS

Theorems

Exponentiation rules

Suitable Grade Level

Grades 9-11