Let's solve the problem step by step: ### **Step 1: Identify the units to convert.** The expression is: \[ 32 \left( \frac{\mu g}{dL} \right) \times \left[ \text{?} \right] = \frac{g}{mL} \] We need to convert **micrograms per deciliter (\(\mu g/dL\))** to **grams per milliliter (\(g/mL\))**. **Explanation:** We need to convert both the numerator (from \(\mu g\) to \(g\)) and the denominator (from \(dL\) to \(mL\)). --- ### **Step 2: Find the appropriate conversion factors.** - \(1 \, \mu g = 10^{-6} \, g\) - \(1 \, dL = 100 \, mL\) So, the conversion factor for \(\mu g\) to \(g\) is \(\frac{10^{-6} \, g}{1 \, \mu g}\). The conversion factor for \(dL\) to \(mL\) is \(\frac{1 \, dL}{100 \, mL}\). **Explanation:** By multiplying by these fractions, the units will cancel appropriately to leave \(g/mL\). --- ### **Step 3: Set up the complete conversion factor.** \[ \left( \frac{10^{-6} \, g}{1 \, \mu g} \right) \times \left( \frac{1 \, dL}{100 \, mL} \right) \] **Explanation:** - The first fraction converts micrograms to grams. - The second fraction converts deciliters to milliliters (in the denominator). --- ## **Final Answer:** \[ \boxed{ \left( \frac{10^{-6} \, g}{1 \, \mu g} \right) \times \left( \frac{1 \, dL}{100 \, mL} \right) } \] Multiply \(32 \left( \frac{\mu g}{dL} \right)\) by this expression to get the answer in \(\frac{g}{mL}\).

The concept used involves unit conversion by multiplying the given measurement by appropriate conversion factors that cancel out original units and convert them into desired units. This process relies on understanding basic metric equivalents, such as micrograms to grams and deciliters to milliliters, ensuring the units are properly aligned for accurate conversion.

The concept used in answering this question is **unit conversion** through the application of **conversion factors**. It involves identifying the initial units (micrograms per deciliter) and the target units (grams per milliliter), then multiplying by fractions that represent the relationships between these units. This process relies on understanding metric equivalents, such as \(1\, \mu g = 10^{-6}\, g\) and \(1\, dL = 100\, mL\). By carefully selecting and multiplying these conversion factors, the units cancel appropriately, allowing the original measurement to be expressed accurately in the desired units. This method ensures precise and systematic conversion between different measurement units in scientific calculations.

Let's proceed with the calculations step by step: ### **Step 1: Write the given expression** Given: \[ 32 \left( \frac{\mu g}{dL} \right) \] We want to convert \(\frac{\mu g}{dL}\) to \(\frac{g}{mL}\). --- ### **Step 2: Write the conversion factors** - \(1\, \mu g = 10^{-6}\, g\) - \(1\, dL = 100\, mL\) Expressed as fractions: \[ \frac{10^{-6}\, g}{1\, \mu g} \quad \text{and} \quad \frac{1\, dL}{100\, mL} \] ### **Step 3: Set up the complete conversion** Multiply the original measurement by the conversion factors: \[ 32 \times \left( \frac{10^{-6}\, g}{1\, \mu g} \right) \times \left( \frac{1\, dL}{100\, mL} \right) \] --- ### **Step 4: Simplify the expression** Combine the constants: \[ 32 \times 10^{-6} \times \frac{1}{100} \] Calculate step-by-step: \[ 32 \times 10^{-6} = 3.2 \times 10^{-5} \] Then divide by 100: \[ \frac{3.2 \times 10^{-5}}{100} = 3.2 \times 10^{-5} \times 10^{-2} = 3.2 \times 10^{-7} \] ### **Final calculation:** \[ \boxed{ 32 \left( \frac{\mu g}{dL} \right) = 3.2 \times 10^{-7} \frac{g}{mL} } \] --- ## **Final Answer:** \[ \boxed{ \boxed{ 3.2 \times 10^{-7} \, \frac{g}{mL} } } \] This is the equivalent concentration in grams per milliliter.