Problem 54 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ \sqrt{25} $$

Step-by-Step Solution

Answer

The result is 5

Step 1: Identify the Expression

We need to evaluate or simplify: \(\sqrt{25}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

The result is 5

Key Concepts

Perfect Squares

Understanding perfect squares is integral to mastering basic algebra and simplifying square root expressions. A perfect square is the product of a numeral multiplied by itself. For instance, the number 4 is a perfect square because it's 2 times 2, and the number 25 is also a perfect square because it's the result of 5 times 5.

Identifying perfect squares can make square root calculations much simpler. For numbers like 1, 4, 9, 16, 25, 36, and so forth, the square root is an integer. The concept of perfect squares extends beyond the basic examples, and recognizing them helps in evaluating square roots without the need for a calculator. Here's a quick tip: if the number ends in 6, it's last digit when squared will also end in 6, similar patterns can be spotted for other digits as well.

Square Root Calculations

When calculating square roots, understanding the principal square root is essential. This value represents the positive square root of a number. For positive numbers, this is denoted by \( \sqrt{x} \) and doesn't contain a plus-minus symbol, which is usually associated with solving equations that include squaring variables.

In our exercise, evaluating \( \sqrt{25} \) is straightforward because 25 is a perfect square, and its positive square root is 5. To calculate the square root of numbers that are not perfect squares, you might need to use the long division method or a calculator, which typically follows algorithms to factor the number down and find the closest possible value to the real square root.

Approximation Methods

When we're dealing with numbers that are not perfect squares, approximation methods come into play. These techniques allow us to estimate the square root of a number that doesn't have a neat, whole number as its square root. One common method is finding the nearest perfect squares that are less than and greater than the given number and then using interpolation to estimate the root.

For more precision, the Babylonian method, or Hero's method, is a formula that iteratively improves the estimate of the square root with each calculation. Another technique involves using the power series expansion or Newton's method for an even more accurate approximation. However, for most schoolwork purposes, if a non-perfect square root needs to be estimated, a calculator is the most efficient tool, rounding to the required degree of precision like the nearest hundredth.