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Express the given expression without logs, in simplest form. Assume all variables represent positive values. (11^(log_(11)(7sqrtw)-log_(11)(9y^(2)))) Answer:

Express the given expression without logs, in simplest form. Assume all variables represent positive values.\newline(11log⁡11(7w)−log⁡11(9y2)) \left(11^{\log _{11}(7 \sqrt{w})-\log _{11}\left(9 y^{2}\right)}\right) \newlineAnswer:

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Q. Express the given expression without logs, in simplest form. Assume all variables represent positive values.\newline(11log⁡11(7w)−log⁡11(9y2)) \left(11^{\log _{11}(7 \sqrt{w})-\log _{11}\left(9 y^{2}\right)}\right) \newlineAnswer:
  1. Recognize Properties: Recognize the properties of logarithms that will be used to simplify the expression.\newlineThe expression involves the laws of logarithms, specifically the quotient rule which states that log⁡b(a)−log⁡b(c)=log⁡b(ac)\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right). We will use this property to combine the logarithms in the exponent.
  2. Apply Quotient Rule: Apply the quotient rule to the logarithms in the exponent.\newlineUsing the quotient rule, we can rewrite log⁡11(7w)−log⁡11(9y2)\log_{11}(7\sqrt{w}) - \log_{11}(9y^2) as log⁡11(7w9y2)\log_{11}\left(\frac{7\sqrt{w}}{9y^2}\right).
  3. Simplify Expression: Simplify the expression inside the logarithm.\newlineThe expression inside the logarithm can be simplified by recognizing that w\sqrt{w} is w1/2w^{1/2}. So, we have log⁡11(7w1/29y2)\log_{11}\left(\frac{7w^{1/2}}{9y^2}\right).
  4. Apply Power Rule: Apply the power rule of exponents to the base 1111 with the logarithm as the exponent.\newlineAccording to the power rule, blog⁡b(x)=xb^{\log_b(x)} = x for any base bb and positive xx. Therefore, 11log⁡11(7w1/29y2)11^{\log_{11}\left(\frac{7w^{1/2}}{9y^2}\right)} simplifies to 7w1/29y2\frac{7w^{1/2}}{9y^2}.
  5. Check Simplified Form: Check if the expression is in its simplest form.\newlineThe expression (7w12)/(9y2)(7w^{\frac{1}{2}})/(9y^2) is already in its simplest form, as there are no common factors that can be cancelled out between the numerator and the denominator.

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