Answer: [D]: 5 .
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Explanation:
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The problem given is:
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25^(⅓) * 25^(⅙) = ?
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Note the following property, or rule, of exponents:
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aᵐ * aⁿ = a⁽ ᵐ⁺ⁿ ⁾ ;
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As such:
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25^(⅓) * 25^(⅙) = 25^[ (⅓) + (⅙) ] ;
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Now, let us add:
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(⅓) + (⅙) = ??
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(1/3) = ?/6 ?
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(1/3) = (1*2)/(3*2) ; {since "6 ÷ 3 = 2"} ;
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→ (1/3) = (1*2)/(3*2) = 2/6 ;
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Now, rewrite the entire problem:
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→ 25^(⅓) * 25^(⅙) = 25^[ (⅓) + (⅙) ] ;
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= 25^ [(2/6)+ (⅙) ] ;
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→ [ (2/6) + (⅙) ] = 3/6 ;
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→ "3/6" can be further reduced:
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→ "3/6" = (3÷3)/(6÷3) = 1/2 ; that is: "½" .
→ Alternately, by inspection, we know that: "3/6" = "½" ; ;
since: "3" is one-half of "6" .
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So, let us rewrite the problem:
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→ 25^(⅓) * 25^(⅙) = 25^[ (⅓) + (⅙) ] ;
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= 25^[(2/6) + (⅙) ] ;
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= 25^(3/6) ;
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= 25^(½) ;
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→ Note the following rule, or property of exponents:
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→ ⁿ√(aᵐ) = [ ⁿ√(a) ] ᵐ = a^(m/n) ;
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So, since:
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→ ⁿ√(aᵐ) = a^(m/n) ;
then: ↔ a^(m/n) = ⁿ√(aᵐ) ;
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And as such:
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→ 25^(½) = ²√(25¹) ;
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So, we have:
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→ ²√(25¹) ; Note: for "square roots", the "superscript 2" is implied, so we can eliminate that "superscript 2"; and rewrite as:
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→ √(25¹) ;
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Note: 25¹ = 25; so:
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→ √(25¹) = √25 ;
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→ √25 = 5 ; which is our answer; which corresponds to:
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Answer choice: [D]: 5 .
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