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What is the expression for f(x) when we rewrite {:[((1)/(32))^(x)*((1)/(2))^(9x-5)" as "((1)/(2))^(f(x))],[" ? "],[f(x)=]:}

What is the expression for f(x) f(x) when we rewrite\newline(132)x⋅(12)9x−5 as (12)f(x)f(x)=□ \begin{array}{l} \left(\frac{1}{32}\right)^{x} \cdot\left(\frac{1}{2}\right)^{9 x-5} \text { as }\left(\frac{1}{2}\right)^{f(x)} \\ f(x)=\square \end{array}

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Full solution

Q. What is the expression for f(x) f(x) when we rewrite\newline(132)x⋅(12)9x−5 as (12)f(x)f(x)=□ \begin{array}{l} \left(\frac{1}{32}\right)^{x} \cdot\left(\frac{1}{2}\right)^{9 x-5} \text { as }\left(\frac{1}{2}\right)^{f(x)} \\ f(x)=\square \end{array}
  1. Simplify given expression: We need to express the given function in the form of ((12)f(x))((\frac{1}{2})^{f(x)}). To do this, we will first simplify the given expression using the properties of exponents.\newlineThe given expression is ((132)x⋅(12)9x−5)((\frac{1}{32})^x \cdot (\frac{1}{2})^{9x-5}).\newlineWe know that (132)(\frac{1}{32}) can be written as (12)5(\frac{1}{2})^5 because 25=322^5 = 32.\newlineSo, ((132)x)((\frac{1}{32})^x) can be rewritten as ((12)5)x((\frac{1}{2})^5)^x.\newlineUsing the property of exponents (am)n=am⋅n(a^{m})^n = a^{m\cdot n}, we get ((12)5⋅x)((\frac{1}{2})^{5\cdot x}).\newlineNow, the expression becomes ((12)5⋅x⋅(12)9x−5)((\frac{1}{2})^{5\cdot x} \cdot (\frac{1}{2})^{9x-5}).
  2. Combine exponents: Next, we will combine the exponents of the same base using the property am×an=am+na^m \times a^n = a^{m+n}. So, we combine the exponents of (1/2)(1/2) by adding them together: 5x+(9x−5)5x + (9x-5). This simplifies to 5x+9x−55x + 9x - 5. Combining like terms, we get 14x−514x - 5. Therefore, the expression can now be written as (1/2)14x−5(1/2)^{14x - 5}.
  3. Express as a single power: We have now expressed the original expression as a single power of (12)(\frac{1}{2}). This means that f(x)=14x−5f(x) = 14x - 5.

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