Problem 9 - Algebra and Trigonometry

Question

Evaluate each expression indicate that the root is not a real number. $$ \sqrt{25}-\sqrt{16} $$

Step-by-Step Solution

Answer

The result of the expression \(\sqrt{25}-\sqrt{16}\) is 1.

Step 1: Calculate the Square Root of 25

The first operation to perform is the square root of 25. The square root of 25 is a number which, when multiplied by itself, gives 25. The square root of 25 is 5 because \(5^2 = 25\).

Step 2: Calculate the Square Root of 16

Next, the square root of 16 has to be calculated. The square root of 16 is a number which, when multiplied by itself, gives 16. The square root of 16 is 4 because \(4^2 = 16\).

Step 3: Subtract the Results

The final step is to subtract the square root of 16 from the square root of 25. That is, \(5 - 4 = 1\).

Key Concepts

Square Root Calculation

Understanding how to calculate square roots is fundamental in mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, when you're asked to evaluate \(\sqrt{25}\), you need to think of the number that times itself equals 25. This process sometimes requires memorization of squares for numbers 1 through 12, as these are often used in fundamental mathematical computations.

The process of finding a square root can often be simplified by identifying perfect squares. Perfect squares are the squares of integer numbers, like 1, 4, 9, 16, and so on. Recognizing these can save you time, as the square root of a perfect square is always an integer. In our exercise, both 25 and 16 are perfect squares, which is why their square roots are easily found to be 5 and 4, respectively.

Simplifying Radical Expressions

At times you'll come across expressions like \(\sqrt{50}\) or \(\sqrt{72}\), which contain square roots that are not as straightforward as perfect squares. These expressions require simplification by breaking them down into their prime factors. For example, the number 50 can be factored into 2 and 25, where 25 is a perfect square. Thus, \(\sqrt{50}\) can be simplified to \(\sqrt{2 \times 25}\) or \(5\sqrt{2}\).

This process helps transform radical expressions into their simplest form, making subsequent arithmetic operations more manageable. To optimize learning, you should practice simplifying a variety of radical expressions. This builds familiarity with prime factors and the relationship between multiplication and square roots.

Arithmetic Operations with Radicals

Performing arithmetic operations with radicals, such as addition, subtraction, multiplication, and division, requires a clear understanding of radical rules. You can add or subtract radicals only if they are like terms, which means they have the same radical part. For example, the expression \(3\sqrt{2} + 2\sqrt{2}\) simplifies to \(5\sqrt{2}\), because the radicals are like terms.

In contrast, the expression from our exercise \(\sqrt{25} - \sqrt{16}\) involves two different radicals, which after calculation become simple integers that can be easily subtracted from each other. However, when simplifying expressions with unlike radicals, you first need to evaluate the square roots, if possible, and then perform the arithmetic operation. Always remember that operations involving radicals follow the usual algebraic rules of operation order, which means you'll often need to do any multiplication or division before addition or subtraction.