Find the square root of the following surd: 8 - 3√7 .

To find the square root of the surd \(8 - 3\sqrt{7}\), we can express it in a simpler form. Let's denote the square root as \(x\), so we have:

Setting Up the Equation

We start with the equation:

x = √(8 - 3√7)

Assuming a Form

We can assume that the square root can be expressed as:

x = √a - √b

where \(a\) and \(b\) are positive numbers we need to determine.

Squaring Both Sides

Next, we square both sides:

x² = a + b - 2√(ab)

Setting this equal to \(8 - 3√7\), we can match the rational and irrational parts:

Matching Terms

• Rational part: \(a + b = 8\)
• Irrational part: \(-2√(ab) = -3√7\)

Solving the Irrational Part

From the irrational part, we can simplify:

2√(ab) = 3√7

Dividing both sides by 2 gives:

√(ab) = (3/2)√7

Squaring both sides results in:

ab = (9/4) * 7 = 63/4

Forming a System of Equations

Now we have two equations:

• 1: \(a + b = 8\)
• 2: \(ab = 63/4\)

Using the Quadratic Formula

We can express \(b\) in terms of \(a\) from the first equation:

b = 8 - a

Substituting this into the second equation gives:

a(8 - a) = 63/4

This simplifies to:

4a(8 - a) = 63

or

32a - 4a² = 63

Rearranging leads to:

4a² - 32a + 63 = 0

Finding the Roots

Using the quadratic formula \(a = \frac{-b ± √(b² - 4ac)}{2a}\), we substitute:

a = 4, b = -32, c = 63

Calculating the discriminant:

(-32)² - 4(4)(63) = 1024 - 1008 = 16

Now, substituting back into the formula:

a = \frac{32 ± 4}{8}

This gives us:

• a = 4.5
• b = 3.5

Final Result

Thus, the square root of \(8 - 3√7\) can be expressed as:

√(8 - 3√7) = √(4.5) - √(3.5)

In conclusion, the square root of the surd \(8 - 3√7\) is approximately:

√(4.5) - √(3.5)