`(sqrt(288))/(sqrt (126))`=?

Options

A. `(sqrt3)/(2)`
B. `(3) sqrt2)`
C. `3/2`
D. `1.49`

Answer

To solve the expression \((\sqrt{288})/(\sqrt{126})\), we can follow these steps:

### Step 1: Prime Factorization
First, we will find the prime factorization of both numbers, 288 and 126.

**For 288:**
- Divide by 2: \(288 \div 2 = 144\)
- Divide by 2: \(144 \div 2 = 72\)
- Divide by 2: \(72 \div 2 = 36\)
- Divide by 2: \(36 \div 2 = 18\)
- Divide by 2: \(18 \div 2 = 9\)
- Divide by 3: \(9 \div 3 = 3\)
- Divide by 3: \(3 \div 3 = 1\)

So, the prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

**For 126:**
- Divide by 2: \(126 \div 2 = 63\)
- Divide by 3: \(63 \div 3 = 21\)
- Divide by 3: \(21 \div 3 = 7\)
- Divide by 7: \(7 \div 7 = 1\)

So, the prime factorization of 126 is:
\[ 126 = 2^1 \times 3^2 \times 7^1 \]

### Step 2: Substitute the Prime Factors into the Square Roots
Now we can substitute the prime factors back into the square roots:
\[ \sqrt{288} = \sqrt{2^5 \times 3^2} \quad \text{and} \quad \sqrt{126} = \sqrt{2^1 \times 3^2 \times 7^1} \]

### Step 3: Simplify the Expression
Now we can write the expression as:
\[ \frac{\sqrt{288}}{\sqrt{126}} = \frac{\sqrt{2^5 \times 3^2}}{\sqrt{2^1 \times 3^2 \times 7^1}} = \frac{\sqrt{2^5} \cdot \sqrt{3^2}}{\sqrt{2^1} \cdot \sqrt{3^2} \cdot \sqrt{7^1}} \]

### Step 4: Simplify the Square Roots
Now we can simplify the square roots:
- \(\sqrt{2^5} = \sqrt{(2^4 \cdot 2)} = 2^2 \cdot \sqrt{2} = 4\sqrt{2}\)
- \(\sqrt{3^2} = 3\)
- \(\sqrt{2^1} = \sqrt{2}\)
- \(\sqrt{7^1} = \sqrt{7}\)

Substituting back, we have:
\[ \frac{4\sqrt{2} \cdot 3}{\sqrt{2} \cdot 3 \cdot \sqrt{7}} = \frac{12\sqrt{2}}{\sqrt{2} \cdot 3 \cdot \sqrt{7}} \]

### Step 5: Cancel Common Terms
Now we can cancel \(\sqrt{2}\) and \(3\):
\[ = \frac{12}{3\sqrt{7}} = \frac{4}{\sqrt{7}} \]

### Step 6: Rationalize the Denominator (Optional)
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{7}\):
\[ \frac{4\sqrt{7}}{7} \]

### Final Answer
Thus, the final answer is:
\[ \frac{4\sqrt{7}}{7} \]

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