Express each of the following ratios in simplest form :
`8ab^(2) : 6a^(2) b `

Answer

To express the ratio \(8ab^2 : 6a^2b\) in its simplest form, we can follow these steps:

### Step 1: Write the ratio in fraction form
We start by writing the given ratio as a fraction:
\[ \frac{8ab^2}{6a^2b} \]

### Step 2: Factor the numerator and the denominator
Next, we factor both the numerator and the denominator:
- The numerator \(8ab^2\) can be factored as:
\[ 8ab^2 = 2 \times 2 \times 2 \times a \times b \times b = 2^3 \times a \times b^2 \]
- The denominator \(6a^2b\) can be factored as:
\[ 6a^2b = 2 \times 3 \times a \times a \times b = 2 \times 3 \times a^2 \times b \]

### Step 3: Write the fraction with the factors
Now we can rewrite the fraction using these factors:
\[ \frac{2^3 \times a \times b^2}{2 \times 3 \times a^2 \times b} \]

### Step 4: Cancel the common factors
We can cancel the common factors in the numerator and the denominator:
- The \(2\) in the denominator cancels with one \(2\) in the numerator.
- One \(a\) in the numerator cancels with one \(a\) in the denominator.
- One \(b\) in the denominator cancels with one \(b\) in the numerator.

After canceling, we have:
\[ \frac{2^2 \times b}{3 \times a} = \frac{4b}{3a} \]

### Step 5: Write the simplest form of the ratio
Thus, the simplest form of the ratio \(8ab^2 : 6a^2b\) is:
\[ 4b : 3a \]

### Final Answer
The simplest form of the ratio is:
\[ 4b : 3a \]

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