If a, b, and c are positive numbers such that `sqrt((a)/(b))=8c and ac=b`, what is the value of c?

Answer

To solve the problem, we have two equations involving the variables \( a \), \( b \), and \( c \):

1. \(\sqrt{\frac{a}{b}} = 8c\)
2. \(ac = b\)

We need to find the value of \( c \).

**Step 1: Substitute \( b \) from the second equation into the first equation.**

From the second equation \( ac = b \), we can express \( b \) in terms of \( a \) and \( c \):

\[ b = ac \]

Now, substitute \( b \) into the first equation:

\[ \sqrt{\frac{a}{ac}} = 8c \]

**Step 2: Simplify the left side of the equation.**

The left side simplifies as follows:

\[ \sqrt{\frac{a}{ac}} = \sqrt{\frac{1}{c}} = \frac{1}{\sqrt{c}} \]

So the equation now looks like:

\[ \frac{1}{\sqrt{c}} = 8c \]

**Step 3: Rearrange the equation.**

To eliminate the fraction, we can multiply both sides by \(\sqrt{c}\):

\[ 1 = 8c\sqrt{c} \]

**Step 4: Express \(\sqrt{c}\) in terms of \( c \).**

We know that \(\sqrt{c} = c^{1/2}\), so we can rewrite the equation as:

\[ 1 = 8c^{3/2} \]

**Step 5: Solve for \( c \).**

Now, we can solve for \( c \):

\[ c^{3/2} = \frac{1}{8} \]

To isolate \( c \), we raise both sides to the power of \(\frac{2}{3}\):

\[ c = \left(\frac{1}{8}\right)^{\frac{2}{3}} \]

**Step 6: Simplify the expression.**

Calculating \(\left(\frac{1}{8}\right)^{\frac{2}{3}}\):

\[ \frac{1}{8} = \frac{1}{2^3} \implies \left(\frac{1}{2^3}\right)^{\frac{2}{3}} = \frac{1^{2/3}}{2^{2}} = \frac{1}{4} \]

Thus, we have:

\[ c = \frac{1}{4} \]

**Final Answer:**

The value of \( c \) is \(\frac{1}{4}\).

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