Problem 25 - Algebra 1

Question

Simplify the expression. $$\sqrt{72}-\sqrt{18}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt{72} - \sqrt{18} \) is 3\( \sqrt{2} \).

Step 1: Break down the square roots

Break down the square roots into a product of square roots. One factor should be the greatest perfect square, while the other factor should be the remaining factor. In this case, \( \sqrt{72} \) can be written as \( \sqrt{36} \cdot \sqrt{2} \) and \( \sqrt{18} \) can be written as \( \sqrt{9} \cdot \sqrt{2} \). So it becomes: \( \sqrt{36} \cdot \sqrt{2} - \sqrt{9} \cdot \sqrt{2} \)

Step 2: Simplify the square roots of perfect squares

Next, simplify the square roots of perfect squares. The square root of 36 is 6 and the square root of 9 is 3. This reduces the expression to: \( 6 \cdot \sqrt{2} - 3 \cdot \sqrt{2} \)

Step 3: Combine like terms

These expressions can be combined because they're like terms. Subtracting 3\( \sqrt{2} \) from 6\( \sqrt{2} \) leaves 3\( \sqrt{2()} \). So, the simplified form of the given expression is 3\( \sqrt{2} \).

Key Concepts

Perfect Squares

Understanding perfect squares is essential in simplifying square roots. A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For instance, 1, 4, 9, 16, and 25 are all perfect squares because they can be written as 1×1, 2×2, 3×3, 4×4, and 5×5, respectively.

When simplifying square roots like in the equation \( \sqrt{72} - \sqrt{18} \), identifying the largest perfect square that is a factor of the number under the square root sign is crucial. To exemplify, 72 can be factored into 36 and 2, where 36 is the greatest perfect square. This process simplifies the square root of 72 to \( \sqrt{36} \cdot \sqrt{2} \), which then becomes 6\( \sqrt{2} \) upon simplification because \( \sqrt{36} \) equals 6.

Radicals

Radicals, otherwise known as roots, are the inverse operation to exponentiation. The square root symbol \( \sqrt{} \) is a common example, and it specifically denotes the principal square root. When simplifying radicals without a calculator, the objective is to find the simplest form that removes the radical sign.

A step-by-step method to simplify radicals involves factoring the number under the radical into perfect squares as shown in the previous section. After factoring, you extract the square root of the perfect squares, turning them into whole numbers. For the number 18, you can rewrite \( \sqrt{18} \) as \( \sqrt{9} \cdot \sqrt{2} \), where \( \sqrt{9} \) is 3. This approach makes the process of simplifying radicals methodical and precise.

Combine Like Terms

Combining like terms is a fundamental part of simplifying expressions in algebra. Terms that are 'like' have the same variable raised to the same power. With radicals, like terms have the same radical part. For instance, in the expression 6\( \sqrt{2} \) - 3\( \sqrt{2} \), both terms have the \( \sqrt{2} \) radical.

When you combine like terms, you only add or subtract the coefficients (the numbers in front of the radicals). In the given example, 6 minus 3 gives us 3, so combining these like terms gives us 3\( \sqrt{2} \). It is similar to saying that if you have six apples and you lose three, you're left with three apples. Hence, knowledge of how to combine like terms is crucial in reaching a simplified form in algebraic expressions involving radicals.