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Solve the exponential equation for x. {:[(64^(7x-1))/(4^(2x+3))=4^(9x-4)],[x=◻]:}

Solve the exponential equation for x x .\newline647x−142x+3=49x−4x=□ \begin{array}{l} \frac{64^{7 x-1}}{4^{2 x+3}}=4^{9 x-4} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline647x−142x+3=49x−4x=□ \begin{array}{l} \frac{64^{7 x-1}}{4^{2 x+3}}=4^{9 x-4} \\ x=\square \end{array}
  1. Simplify Exponential Equation: Simplify the given exponential equation using the properties of exponents.\newlineGiven equation: 64(7x−1)4(2x+3)=4(9x−4)\frac{64^{(7x-1)}}{4^{(2x+3)}} = 4^{(9x-4)}\newlineSince 64=4364 = 4^3, we can rewrite 64(7x−1)64^{(7x-1)} as (43)(7x−1)(4^3)^{(7x-1)}.\newlineNow the equation becomes: (43)(7x−1)4(2x+3)=4(9x−4)\frac{(4^3)^{(7x-1)}}{4^{(2x+3)}} = 4^{(9x-4)}\newlineUsing the property amn=(am)na^{mn} = (a^m)^n, we get: 43(7x−1)=421x−34^{3(7x-1)} = 4^{21x-3}\newlineNow the equation is: 4(21x−3)4(2x+3)=4(9x−4)\frac{4^{(21x-3)}}{4^{(2x+3)}} = 4^{(9x-4)}\newlineUsing the property aman=a(m−n)\frac{a^m}{a^n} = a^{(m-n)}, we get: 4(21x−3−(2x+3))=4(9x−4)4^{(21x-3-(2x+3))} = 4^{(9x-4)}\newlineSimplify the exponent on the left side: 64=4364 = 4^300\newlineThis simplifies to: 64=4364 = 4^311
  2. Rewrite Using Properties: Since the bases are the same, we can set the exponents equal to each other.\newline19x−6=9x−419x - 6 = 9x - 4
  3. Set Exponents Equal: Solve for x by isolating the variable.\newlineSubtract 99x from both sides: 19x−9x−6=9x−9x−419x - 9x - 6 = 9x - 9x - 4\newlineThis simplifies to: 10x−6=−410x - 6 = -4\newlineAdd 66 to both sides: 10x−6+6=−4+610x - 6 + 6 = -4 + 6\newlineThis simplifies to: 10x=210x = 2\newlineDivide both sides by 1010: 10x10=210\frac{10x}{10} = \frac{2}{10}\newlineThis simplifies to: x=15x = \frac{1}{5}

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