Problem 66 - Prealgebra

Question

Compute the exact value of the given expression. \(-1+\sqrt{625}\)

Step-by-Step Solution

Answer

The exact value is 24.

Step 1: Identify the Expression

We are given the expression \(-1 + \sqrt{625}\). The task is to compute its exact value by simplifying it.

Step 2: Evaluate the Square Root

The expression contains \(\sqrt{625}\). Recall that \(625\) is a perfect square. Find its square root by determining the number which, when squared, equals \(625\). We know that \(25 \times 25 = 625\). Thus, \(\sqrt{625} = 25\).

Step 3: Substitute and Simplify

Substitute the evaluated square root back into the expression. The expression becomes \(-1 + 25\). Simplify this by performing the arithmetic operation: \(-1 + 25 = 24\).

Step 4: State the Exact Value

The exact value of the expression \(-1 + \sqrt{625}\) is 24. Confirm that all calculations are correct and conclude the result.

Key Concepts

Understanding Square Roots

A square root is a number that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \). For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). Understanding square roots is essential because they help simplify expressions involving large numbers.
Square roots are commonplace in mathematical equations.

• To find a square root, look for a number that can be multiplied by itself to produce the number under the square root symbol.
• The square root of a perfect square, like 625, is a whole number.

Knowledge of perfect squares aids in identifying the square root quickly, as seen in \( \sqrt{625} = 25 \). Being able to calculate square roots effortlessly is crucial for simplifying complex expressions.

Recognizing Perfect Squares

Perfect squares are numbers that result from multiplying a whole number by itself. They are significant in mathematics because their square roots are always integers. Examples include 16, created by \( 4 \times 4 \), and 36, from \( 6 \times 6 \). Identifying perfect squares allows quick simplification of expressions.
Some key characteristics of perfect squares include:

• They are positive integers.
• Their factors are pairs of the same number, such as \( 25 \times 25 = 625 \).
• Recognizing perfect squares helps in finding square roots without a calculator.

This is handy in solving expressions like \( -1 + \sqrt{625} \), resulting in a simplified value when \( \sqrt{625} \) is calculated to be 25.

Performing Arithmetic Operations

Arithmetic operations form the foundation of mathematics, involving the basic computation methods: addition, subtraction, multiplication, and division. They are used to simplify expressions and solve equations.
When solving an expression with arithmetic operations:

• Identify the numbers and operations present.
• Apply the operations in the correct order, respecting parentheses and other mathematical conventions.

For instance, in \( -1 + \sqrt{625} \), after finding \( \sqrt{625} = 25 \), substitute to get \( -1 + 25 \). Perform the arithmetic operation to find \( 24 \).
Practicing these operations boosts confidence in dealing with more complex problems, ensuring fluency in mathematical calculations.