Math Problem Statement
Simplify
Simplify 4^3 * 32^3 * 8^5
Solution
Let’s simplify the given mathematical expression:
The expression is:
Step 1: Express all terms with the same base
We know that:
- ,
- ,
- .
Rewrite the expression with base 2:
32^3 = (2^5)^3 = 2^{15}, \quad 8^5 = (2^3)^5 = 2^{15}.$$ So the expression becomes: $$2^6 \cdot 2^{15} \cdot 2^{15}.$$ ### Step 2: Combine the powers Using the property of exponents ($$a^m \cdot a^n = a^{m+n}$$): $$2^6 \cdot 2^{15} \cdot 2^{15} = 2^{6 + 15 + 15} = 2^{36}.$$ ### Final Simplified Expression: $$\boxed{2^{36}}$$ Let me know if you'd like further clarification! 😊 --- ### 5 Related Questions: 1. How do you simplify expressions with different bases? 2. What is the value of $$2^{36}$$ as a decimal? 3. How would the simplification change if one of the bases were different, e.g., $$3^3$$? 4. Can you explain the rule $$(a^m)^n = a^{m \cdot n}$$ with examples? 5. What happens if the bases are not powers of the same number? ### Tip: Always break numbers into their prime factorization to simplify expressions involving powers!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponents
Prime Factorization
Properties of Exponents
Formulas
a^m * a^n = a^(m+n)
(a^m)^n = a^(m*n)
Theorems
Laws of Exponents
Suitable Grade Level
Grades 8-10