Answer: " [tex]\frac{8b^4}{3a^1^2}[/tex] " .
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Step-by-step explanation:
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We are asked to simplify the following given expression:
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[tex]\frac{(\frac{2b^3c}{a^3b^2c^2})^4}{6}[/tex] ;
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Let us start with the "numerator" ; which functions as a separate "fraction"—in and of itself—with its own numerator and denominator:
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[tex](\frac{2b^3c}{a^3b^2c^2})^4[/tex] ;
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Note the following "division" property of exponents:
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→ [tex](\frac{a}{b})^c = \frac{a^c}{b^c}[/tex] ; [tex]b\neq 0[/tex] .
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Then:
→ (\frac{2b^3c}{a^3b^2c^2})^4 ;
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= [tex]\frac{(2b^3c)^4}{(a^3b^2c^2)^4}[/tex] ; [tex]a\neq 0[/tex] ; [tex]b\neq 0[/tex] ; [tex]c\neq 0[/tex] .
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Then, start by simplifying "this particular numerator":
→ [tex](2b^3c)^4 = ?[/tex] ;
Note the following "multiplication" property of exponents:
→ [tex](a^b)^c = a^(^b^*^c^)[/tex] ; [tex]b\neq 0[/tex] .
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Then:
→ [tex](2b^3c)^4 = 2^4 * b^(^3^*^4^)(c^4) = (2*2*2*2) * b^1^2 * c^4 = 16 b^1^2c^4[/tex] ;
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The continue by simplifying "this particular denominator":
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→[tex](a^3b^2c^2)^4 = a^(^3^*^4^)b^(^2^*^4^)c^(^2^*^4^) = a^1^2b^8c^8[/tex]
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And we have:
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→ [tex]\frac{16b^1^2c^4}{a^1^2b^8c^8}[/tex] .
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Note the following properties of exponents:
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→ [tex]a^-^b = \frac{1}{a^b}[/tex] ; [tex]a\neq 0[/tex] .
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→ [tex]\frac{a^c}{a^b} = a^(^c^-^b^)[/tex] ; [tex]a \neq 0[/tex] .
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So, we have:
→ \frac{16b^1^2c^4}{a^1^2b^8c^8} ;
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→ 16 ÷ 1 = 16 ; ("1" is the "implied coefficient") ; in the numerator.
→ [tex]a^1^2[/tex] ; stays in the denominator;
→ [tex]\frac{b^1^2}{b^8} = b^(^1^2^-^8^)=b^4[/tex] ; replaces the original: "[tex]b^1^2[/tex] " [in the numerator]; [tex]b^8[/tex] is eliminated in the denominator;
→ [tex]\frac{c^4}{c^8} = c^(^4^-^8^) = c^-^4 = \frac{1}{c^4}[/tex] ; The original "[tex]c^4[/tex] " [in the numerator] is eliminated; the original "[tex]c^8[/tex] " [in the denominator] is replaced with "[tex]c^4[/tex] " . ________________
And the expression is rewritten as:
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→ [tex]\frac{16b^4}{a^1^2c^4}[/tex] ;
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→ Now, from the original given problem, we divide this value by "6" ;
which is the same value we get by multiplying this value by: " [tex]\frac{1}{6}[/tex] " ;
→ as follows:
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→ [tex]\frac{16b^4}{a^1^2c^4} *\frac{1}{6}[/tex] ;
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Considering the "16" and the "6" ; each of these can be divided by "2" ;
Specifically, " (16 ÷2 = 8) " ; and: " (6 ÷ 2 = 3) " .
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So, we can rewrite the expression — by substituting "8" in lieu of the "16" ; and "3" in lieu of the "6" ; as follows:
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→ [tex]\frac{8b^4}{a^1^2} * \frac{1}{3}[/tex] ;
And further simplify;
→ [tex]\frac{8b^4}{a^1^2} * \frac{1}{3} = \frac{(8b^4 * 1)}{(a^1^2*3)} = \frac{8b^4}{3a^1^2} .[/tex]
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Hope this helps!
Best wishes!
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