1/9^(x) / 3^(x) = 27 × (1/81)^x

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Rewrite all terms in the equation with a common base, which is 3. We know that 9 = 3², 27 = 3³, and 81 = 3⁴. Substitute these into the given equation: ((3²)^(2x))/(3^x) = 3³ × ((1)/(3⁴))^(x); Step 2: Apply the exponent rules (a^m)ⁿ = a^(mn) and (1)/(aⁿ) = a⁻ⁿ. (3^(2 × 2x))/(3^x) = 3³ × (3⁻⁴)^(x); (3^(4x))/(3^x) = 3³ × 3^(−4x); Step 3: Apply the exponent rules (a^m)/(aⁿ) = a^(m−n) and a^m × aⁿ = a^(m+n). 3^(4x − x) = 3^(3 + (−4x)); 3^(3x) = 3^(3 − 4x) Step 4: Since the bases are equal, equate the exponents. 3x = 3 − 4x Step 5: Solve for x. Add 4x to both sides of the equation. 3x + 4x = 3; 7x = 3; x = 3/7 The final answer is 3/7.

1/9^(x) / 3^(x) = 27 * (1/81)^x

Answer

Step-by-step solution

Rewrite all terms in the equation with a common base, which is 3.

Apply the exponent rules $(a^m)^n = a^{mn}$ and $\frac{1}{a^n} = a^{-n}$ .

Apply the exponent rules $\frac{a^m}{a^n} = a^{m-n}$ and $a^m \times a^n = a^{m+n}$ .

Since the bases are equal, equate the exponents.

Solve for $x$ .

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