Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt{8 x^{3}}-3 \sqrt{50 x}$$

Short answer

(2x - 15) \sqrt{2x}

Step-by-step solution

Simplify the Radicals

Break down each square root into its prime factors. For \( \sqrt{8x^3} \), we can write it as \( \sqrt{4 \cdot 2 \cdot x^2 \cdot x} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x^2} \cdot \sqrt{x} = 2x \sqrt{2x} \). For \( \sqrt{50x} \), write it as \( \sqrt{25 \cdot 2x} = \sqrt{25} \cdot \sqrt{2x} = 5 \sqrt{2x} \).

Substitute the Simplified Radicals

Replace the original expressions with the simplified forms. The expression becomes \( 2x \sqrt{2x} - 3(5 \sqrt{2x}) = 2x \sqrt{2x} - 15 \sqrt{2x} \).

Combine Like Terms

Combine the like terms, which in this case are \( 2x \sqrt{2x} \) and \( 15 \sqrt{2x} \). This can be written as \( (2x - 15) \sqrt{2x} \).

Key concepts

Radical Expressions

Radical expressions consist of a root, such as a square root or cube root, applied to a number or variable. These expressions can look complicated, but they follow specific rules that make them easier to work with.
Breaking down radicals into simpler forms often involves prime factorization. Remember, when dealing with radicals, it's helpful to express the radicand (the number or expression inside the radical) as a product of factors that are easier to manage.

Like Terms

Like terms in algebra are terms that have the same variable raised to the same power. For example, in the expression given, the terms \(2x \sqrt{2x}\) and \(-15 \sqrt{2x}\) share the same radical \(\sqrt{2x}\). To simplify expressions, always combine like terms. This often involves factoring out the common radical parts and simplifying the coefficients.
Always scan each term closely to identify and separate like terms efficiently.

Prime Factorization

Prime factorization is breaking down a number into a product of its prime factors. For example, \(8\) can be expressed as \(2 \times 2 \times 2\) or \(2^3\), and \(50\) can be written as \(25 \times 2 = 5^2 \times 2\). Prime factorization helps to simplify radicals, especially when splitting them into smaller, more manageable numbers.
This technique is indispensable when simplifying square roots as it allows us to easily take out squares of prime factors from under the radical.

Square Roots

The square root of any number \(n\) is a value that, when multiplied by itself, gives \(n\). For instance, the square root of \(25\) is \(5\), since \(5 \times 5 = 25\).
When simplifying square roots involving variables, look for perfect squares that can be taken out from under the radical, like how \( \sqrt{8x^3} \) was simplified by recognizing \(8\) as \(4 \times 2\) and \(x^3\) as \(x^2 \times x\).
Recognizing and manipulating these square products are key to simplifying expressions containing square roots effectively.