solve for x 6^4x-9=(1/36)^x-4

Identify base and exponent: Identify the base and the exponent on both sides of the equation. On the left side, we have 6^(4x-9), and on the right side, we have (1/36)^(x-4).Recognize 1/36 as 6^-2: Recognize that 1/36 can be written as 6^-2. So, (1/36)^(x-4) can be rewritten as (6^-2)^(x-4).Apply power of power rule: Apply the power of a power rule to the right side of the equation. According to the power of a power rule, (a^m)^n = a^(m*n). So, (6^-2)^(x-4) becomes 6^(-2*(x-4)).Distribute exponent: Distribute the exponent on the right side. 6^(-2*(x-4)) becomes 6^(-2x+8).Set exponents equal: Set the exponents equal to each other since the bases are the same. We now have 6^(4x-9) = 6^(-2x+8). So, 4x - 9 = -2x + 8.Solve for x: Solve for x by combining like terms. Add 2x to both sides to get 4x + 2x = 8 + 9. This simplifies to 6x = 17.Divide by 6: Divide both sides by 6 to solve for x. x = 17 / 6.

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Grade 8

Solve equations with variable exponents

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x

6^{4x-9}=(\frac{1}{36})^{x-4}

Identify base and exponent: Identify the base and the exponent on both sides of the equation. $\newline$ On the left side, we have $6^{4x-9}$ , and on the right side, we have $(1/36)^{x-4}$ .
Recognize $1/36$ as $6^{-2}$ : Recognize that $1/36$ can be written as $6^{-2}$ . So, $(1/36)^{(x-4)}$ can be rewritten as $(6^{-2})^{(x-4)}$ .
Apply power of power rule: Apply the power of a power rule to the right side of the equation. $\newline$ According to the power of a power rule, $(a^m)^n = a^{(m*n)}$ . $\newline$ So, $(6^{-2})^{(x-4)}$ becomes $6^{(-2*(x-4))}$ .
Distribute exponent: Distribute the exponent on the right side. $6^{(-2*(x-4))}$ becomes $6^{(-2x+8)}$ .
Set exponents equal: Set the exponents equal to each other since the bases are the same. $\newline$ We now have $6^{4x-9} = 6^{-2x+8}$ . $\newline$ So, $4x - 9 = -2x + 8$ .
Solve for x: Solve for x by combining like terms. $\newline$ Add $2x$ to both sides to get $4x + 2x = 8 + 9$ . $\newline$ This simplifies to $6x = 17$ .
Divide by $6$ : Divide both sides by $6$ to solve for $x$ . $x = \frac{17}{6}$ .