Math Problem Statement

Simplify the expression \(\frac{2^5 \cdot 2^3}{2^2 \cdot 2^4}\).

Solution

The expression provided in the image is:

25232224\frac{2^5 \cdot 2^3}{2^2 \cdot 2^4}

To simplify this, let's follow the rules of exponents:

  1. Combine the exponents in the numerator and the denominator:

    • In the numerator: 2523=25+3=282^5 \cdot 2^3 = 2^{5+3} = 2^8.
    • In the denominator: 2224=22+4=262^2 \cdot 2^4 = 2^{2+4} = 2^6.
  2. Now, simplify the fraction: 2826=286=22\frac{2^8}{2^6} = 2^{8-6} = 2^2

  3. Calculate 222^2: 22=42^2 = 4

So, the simplified result is 44.

Would you like further details or have any questions?

Here are 5 related questions to explore:

  1. How do you multiply powers with the same base?
  2. What is the rule for dividing powers with the same base?
  3. How would the result change if the exponents in the numerator and denominator were different?
  4. What happens if the base numbers are different in such expressions?
  5. How do negative exponents affect such calculations?

Tip: When multiplying powers with the same base, you add their exponents; when dividing, you subtract the exponents.

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Math Problem Analysis

Mathematical Concepts

Exponents
Multiplication of Powers
Division of Powers

Formulas

Multiplying powers: a^m \cdot a^n = a^{m+n}
Dividing powers: \frac{a^m}{a^n} = a^{m-n}

Theorems

Laws of Exponents

Suitable Grade Level

Grades 6-8