%5E5---2-cdot-(5%5E4)%5E6)-div-5%5E-23-%24.jpeg "$((5^5)^5 - 2 Cdot (5^4)^6) Div 5^ 23 $")
Simplifying the Expression: $((5^5)^5 - 2 \cdot (5^4)^6) \div 5^{23}$
This expression might seem daunting at first, but it can be simplified significantly using the properties of exponents. Let's break it down step by step:
Understanding the Properties of Exponents
- Power of a Power: $(a^m)^n = a^{m \cdot n}$
- Multiplication of Powers: $a^m \cdot a^n = a^{m+n}$
- Division of Powers: $a^m \div a^n = a^{m-n}$
Simplifying the Expression
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Simplify the powers within the parentheses:
- $(5^5)^5 = 5^{5 \cdot 5} = 5^{25}$
- $(5^4)^6 = 5^{4 \cdot 6} = 5^{24}$
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Substitute the simplified powers back into the original expression:
- $((5^5)^5 - 2 \cdot (5^4)^6) \div 5^{23} = (5^{25} - 2 \cdot 5^{24}) \div 5^{23}$
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Factor out a common factor of $5^{24}$ from the numerator:
- $(5^{25} - 2 \cdot 5^{24}) \div 5^{23} = (5^{24}(5 - 2)) \div 5^{23}$
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Apply the division property of exponents:
- $(5^{24}(5 - 2)) \div 5^{23} = 5^{24 - 23} \cdot (5 - 2)$
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Simplify the expression:
- $5^{24 - 23} \cdot (5 - 2) = 5^1 \cdot 3 = 15$