If `a : b = b : c`, then `a^4 : b^4` is equal to:

Options

A. `ac : b^(2)`
B. `a^(2) : c^(2)`
C. `c^(2) : a^(2)`
D. `b^(2) : ac`

Answer

To solve the problem where \( a : b = b : c \), and we need to find \( a^4 : b^4 \), we can follow these steps:

### Step 1: Understand the given ratio
We are given that \( a : b = b : c \). This means that the ratios are equal, and we can express this relationship mathematically.

**Hint:** Recall that if two ratios are equal, the cross products are also equal.

### Step 2: Set up the equation
From the ratio \( a : b = b : c \), we can write it as:
\[ \frac{a}{b} = \frac{b}{c} \]
Cross-multiplying gives us:
\[ a \cdot c = b^2 \]

**Hint:** Cross-multiplication is a useful technique to eliminate fractions in ratios.

### Step 3: Express \( b^2 \) in terms of \( a \) and \( c \)
From the equation \( a \cdot c = b^2 \), we can express \( b^2 \) as:
\[ b^2 = a \cdot c \]

**Hint:** Keep in mind that \( b^2 \) is the square of \( b \) and can be used to find \( b^4 \).

### Step 4: Find \( b^4 \)
To find \( b^4 \), we square \( b^2 \):
\[ b^4 = (b^2)^2 = (a \cdot c)^2 = a^2 \cdot c^2 \]

**Hint:** Remember that squaring a product involves squaring each factor.

### Step 5: Set up the ratio \( a^4 : b^4 \)
Now we can write the ratio \( a^4 : b^4 \):
\[ a^4 : b^4 = a^4 : (a^2 \cdot c^2) \]

**Hint:** You can simplify ratios by dividing both terms by a common factor.

### Step 6: Simplify the ratio
We can simplify the ratio:
\[ a^4 : (a^2 \cdot c^2) = \frac{a^4}{a^2 \cdot c^2} = \frac{a^4}{a^2} : c^2 = a^2 : c^2 \]

**Hint:** When simplifying, remember that \( \frac{a^m}{a^n} = a^{m-n} \).

### Final Answer
Thus, the final answer is:
\[ a^4 : b^4 = a^2 : c^2 \]