Question

find the length of \\(x\\)

a) as an exact value
b) as a decimal to the nearest tenth.

Explanation:

Set up the geometric equation

The total height of the left side is \(8\sqrt{2} + 2\sqrt{12}\). The height of the right step is \(5\sqrt{27} - 4\sqrt{18}\). The segment \(x\) represents the difference between these two heights:

$$ x = (8\sqrt{2} + 2\sqrt{12}) - (5\sqrt{27} - 4\sqrt{18}) $$

Simplify each radical term

Using the Simplifying Radicals knowledge point

$$
LATEXBLOCK0
$$

Substitute and expand the expression

Substitute the simplified radicals back into the equation for \(x\):

$$ x = (8\sqrt{2} + 4\sqrt{3}) - (15\sqrt{3} - 12\sqrt{2}) $$

Distribute the negative sign:

$$ x = 8\sqrt{2} + 4\sqrt{3} - 15\sqrt{3} + 12\sqrt{2} $$

Combine like radical terms

Using the Combining Like Radicals knowledge point

$$
LATEXBLOCK1
$$

Calculate the decimal approximation

Evaluate the exact expression to the nearest tenth:

$$
LATEXBLOCK2
$$

Answer:

a) Exact value:
\(20\sqrt{2} - 11\sqrt{3}\)

b) Decimal value to the nearest tenth:
\(9.2\)