Problem 14 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{625} $$

Step-by-Step Solution

Answer

The square root of 625 is 25.

Step 1: Identify the Expression Type

The expression given is \( \sqrt{625} \), which is a square root problem. Our goal is to find the number that, when multiplied by itself, equals 625.

Step 2: Determine the Perfect Square Root

To solve \( \sqrt{625} \), we need to determine if 625 is a perfect square. A perfect square is a number that has an integer as its square root. We try different integers to find the one which satisfies the condition. After testing, we find that \( 25 \times 25 = 625 \).

Step 3: Verify the Calculation

Double-check the calculation: \( 25 \times 25 = 625 \). This confirms that 625 is a perfect square and its square root is indeed 25.

Step 4: State the Final Answer

Conclude the evaluation of \( \sqrt{625} \). The square root calculation shows that \( \sqrt{625} = 25 \).

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. These numbers have a neat and exact square root, which is also an integer. Understanding perfect squares is important because they simplify the process of taking square roots. If you identify a number as a perfect square, you can easily determine its square root without long calculations.

For example, 625 is a perfect square because there exists an integer, 25, such that when you multiply 25 by itself, you get 625. This can be written as \( 25 \times 25 = 625 \).

• Some common perfect squares include: 1, 4, 9, 16, 25, 36, 49, 64, and 81.
• Recognizing perfect squares can speed up solving many math problems, especially those involving square roots.

Real Numbers

Real numbers encompass a wide variety of numbers, including all rational and irrational numbers. They are useful in evaluating expressions since they include both whole numbers and fractions.

• Rational numbers: Numbers that can be expressed as a fraction where both the numerator and the denominator are integers. Examples include 1/2, 4, and 0.
• Irrational numbers: Numbers that cannot be precisely expressed as simple fractions. Examples include \( \pi \) and \( \sqrt{2} \).

When we evaluate \( \sqrt{625} \), we work within the set of real numbers. The result, 25, is an integer and therefore a rational number, as it can be expressed as 25/1. Understanding where particular numbers fall within the broader set of real numbers allows us to understand their properties better.

Integer Multiplication

Integer multiplication is the mathematical process of finding the product of two whole numbers. When evaluating or solving square root problems, multiplying integers becomes crucial.

Understanding this concept deeply helps in verifying whether a number is a perfect square. When solving \( \sqrt{625} \), you multiply 25 by 25 to check if it results in 625. Here, both 25 and 625 involve integer multiplication.

• Basics of multiplication: It is commutative, meaning \( a \times b = b \times a \).
• Use integer multiplication to find whether numbers are perfect squares and to perform quick calculations.

With integer multiplication, the accuracy of locating square roots and verifying solutions improves, making it a valuable skill in a variety of mathematical applications.