Question

4 power 2x-1- 16
power x-1 =384

Answer 1

Answer:

⇒ $x =\frac{11}{4}$

Step-by-step explanation:

Given :

To find the value of x, if

$4^{2x-1} - 16^{x - 1} = 384$

Solution :

⇒ $4^{2x-1} - 16^{x - 1} = 384$

⇒ $4^{2x-1} - 4^{2(x - 1)} = 384$

⇒ $4^{2x-1} - 4^{2x - 2} = 384$

⇒ $4^{2x-2+1} - 4^{2x - 2} = 384$

⇒  $4(4^{2x-2}) - 4^{2x - 2} = 384$

⇒  $4^{2x-2}(4 - 1) = 384$

⇒  $4^{2x-2}(3) = 384$

⇒ $4^{2x-2} = \frac{384}{3} = 128$

⇒ $4^{2x - 2} = 2^{2(2x-1)} = 128$

⇒ $2^{4x - 4} = 128$

⇒ $2^{4x-4} = 2^7$

⇒ $4x - 4 = 7$

⇒ $4x = 7 + 4 = 11$

⇒ $x =\frac{11}{4}$

Answer 2

Answer:

$4^{2x-1}-16^{x-1}=384\quad :\quad x=\dfrac{11}{4}\quad \left(\mathrm{Decimal}:\quad x=2.75\right)$

Step-by-step explanation:

$4^{2x-1}-16^{x-1}=384$

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$\mathrm{Convert\:}16^{x-1}\mathrm{\:to\:base\:}4$

$16^{x-1}=\left(4^2\right)^{x-1}$

$4^{2x-1}-\left(4^2\right)^{x-1}=384$

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$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(4^2\right)^{x-1}=4^{2\left(x-1\right)}$

$4^{2x-1}-4^{2\left(x-1\right)}=384$

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$\mathrm{Apply\:exponent\:rule}:\quad \:a^{b+c}=a^ba^c$

$4^{2x-1}=4^{2x}\cdot \:4^{-1},\:\space4^{2\left(x-1\right)}=4^{2x}\cdot \:4^{-2}$

$4^{2x}\cdot \:4^{-1}-4^{2x}\cdot \:4^{-2}=384$

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$\mathrm{Factor}\:4^{2x}4^{-1}-4^{2x}4^{-2}$

$4^{2x}4^{-1}-4^{2x}4^{-2}$

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$\mathrm{Factor\:out\:common\:term\:}4^{2x}$

$=4^{2x}\left(4^{-1}-4^{-2}\right)$

$4^{2x}\left(4^{-1}-4^{-2}\right)=384$

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$\mathrm{Divide\:both\:sides\:by\:}4^{-1}-4^{-2}$

$\dfrac{4^{2x}\left(4^{-1}-4^{-2}\right)}{4^{-1}-4^{-2}}=\dfrac{384}{4^{-1}-4^{-2}}$

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Simplify;

$4^{2x}=\dfrac{384}{4^{-1}-4^{-2}}$

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$\mathrm{Simplify\:}\dfrac{384}{4^{-1}-4^{-2}}$

$\dfrac{384}{4^{-1}-4^{-2}}$

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$\mathrm{Simplify}\:4^{-1}-4^{-2}$

$4^{-1}-4^{-2}$

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$\mathrm{Apply\:exponent\:rule}:\quad \:a^{-1}=\dfrac{1}{a}$

$4^{-1}=\dfrac{1}{4}$

$=\dfrac{1}{4}-4^{-2}$

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$\mathrm{Apply\:exponent\:rule}:\quad \:a^{-b}=\dfrac{1}{a^b}$

$4^{-2}=\dfrac{1}{4^2}$

$=\dfrac{1}{4}-\dfrac{1}{4^2}$

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$=\dfrac{384}{\dfrac{1}{4}-\dfrac{1}{4^2}}$

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$\mathrm{Join}\:\dfrac{1}{4}-\dfrac{1}{4^2}:\quad \dfrac{3}{16}$

$=\dfrac{384}{\dfrac{3}{16}}$

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$\mathrm{Apply\:the\:fraction\:rule}:\quad \dfrac{a}{\dfrac{b}{c}}=\dfrac{a\cdot \:c}{b}$

$=\dfrac{384\cdot \:16}{3}$

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$\mathrm{Multiply\:the\:numbers:}\:384\cdot \:16=6144$

$=\dfrac{6144}{3}$

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$\mathrm{Divide\:the\:numbers:}\:\dfrac{6144}{3}=2048$

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$4^{2x}=2048$

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$\mathrm{Convert\:}4^{2x}\mathrm{\:to\:base\:}2$

$4^{2x}=\left(2^2\right)^{2x}$

$\left(2^2\right)^{2x}=2048$

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$\mathrm{Convert\:}2048\mathrm{\:to\:base\:}2$

$2048=2^{11}$

$\left(2^2\right)^{2x}=2^{11}$

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$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(2^2\right)^{2x}=2^{2\cdot \:2x}$

$2^{2\cdot \:2x}=2^{11}$

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$\mathrm{If\:}a^{f\left(x\right)}=a^{g\left(x\right)}\mathrm{,\:then\:}f\left(x\right)=g\left(x\right)$

$2\cdot \:2x=11$

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Simplify;

$4x=11$

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$\mathrm{Divide\:both\:sides\:by\:}4$

$\dfrac{4x}{4}=\dfrac{11}{4}$

$\mathrm{Simplify}$

$x=\dfrac{11}{4}$

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$\bold{x=\dfrac{11}{4}}$