Problem 24 - Algebra 1

Question

Simplify the expression. $$\sqrt{80}-\sqrt{45}$$

Step-by-Step Solution

Answer

\( \sqrt{80} - \sqrt{45} = \sqrt{5} \)

Step 1: Simplification of Square Root of 80

The square root of 80 can be simplified by expressing it as the product of the square root of 16 (a perfect square) and the square root of 5, i.e. \( \sqrt{80} = \sqrt{16 \cdot 5} \). According to the properties of square roots, this expression can be simplified further to \( 4\sqrt{5} \). That is because the square root of 16 equals 4.

Step 2: Simplification of Square Root of 45

The square root of 45 can be simplified by expressing it as the product of the square root of 9 (a perfect square) and the square root of 5, i.e. \( \sqrt{45} = \sqrt{9 \cdot 5} \). According to the properties of square roots, this expression can be simplified further to \( 3\sqrt{5} \). That is because the square root of 9 equals 3.

Step 3: Subtract the Simplified Square Roots

The final step is to subtract \( 3\sqrt{5} \) from \( 4\sqrt{5} \). The result is \( \sqrt{5} \), which is the simplified form of the original expression.

Key Concepts

Understanding Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because \(4 \times 4 = 16\). The notation for a square root is a radical symbol \(\sqrt{\cdot}\), and each number can have two square roots: a positive and a negative.

• In common usage, the square root refers to the positive root, also known as the principal square root.
• Square roots are used in various mathematical procedures, especially in solving quadratic equations.

To simplify a square root, we look for perfect square factors. This helps us express the square root in its simplest form, which is exactly what we'll discuss next.

Perfect Squares: The Key to Simplification

A perfect square is an integer that is the square of another integer. For example, 9, 16, 25 are perfect squares because they are respectively \(3^2\), \(4^2\), and \(5^2\). Recognizing perfect squares is crucial when simplifying expressions involving square roots.

• Perfect squares can be used to break down larger numbers into simpler components within a square root.
• This is done by expressing the original number as a product of a perfect square and another integer.

For instance, to simplify \(\sqrt{80}\), we observe that 80 can be rewritten as \(16 \times 5\), where 16 is a perfect square. Thus, \(\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}\). This method allows us to reduce the complexity of square roots by isolating and simplifying the perfect square component.

Properties of Square Roots

Square roots come with several useful properties that make simplifying expressions more manageable.

• One basic property is that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This property helps break down complicated square roots into simpler, more manageable parts.
• For instance, when simplifying \(\sqrt{45}\), we can express it as \(\sqrt{9 \times 5}\), which simplifies to \(3\sqrt{5}\) because \(\sqrt{9} = 3\).
• Additionally, addition and subtraction of square roots follow simple arithmetic rules if the radicals (numbers under the square root) are the same. This allows expressions like \(4\sqrt{5} - 3\sqrt{5}\) to be simplified to \(\sqrt{5}\).

Mastering these properties helps in efficiently and effectively handling mathematical problems involving radicals, especially in algebraic operations.