Problem 32 - Algebra 1

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$\sqrt{64}$$

Step-by-Step Solution

Answer

The square root of 64 is 8.

Step 1: Understanding the Term

Firstly, it’s important to know what square root means. The square root of a number x is a number that, when multiplied by itself, equals x. In this case, we are looking to find a number which, when multiplied by itself, equals 64.

Step 2: Finding the Square Root of 64

We are solving for \( \sqrt{64} \). The number that multiplies by itself to give 64 is 8. This is because 8*8 equals 64.

Key Concepts

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value or set of values. They can include operations like addition, subtraction, multiplication, and division. In the context of evaluating square roots, a mathematical expression such as \(\sqrt{64}\) represents the operation of finding a number which, when squared, will equal 64.

• The square root symbol \(\sqrt{ }\) is used to denote this operation.
• A square root expression is often simplified to its most compact form—here, simply the number 8.

Expressing values as square roots can be useful in solving equations, understanding geometry, and analyzing data.

Exact Values

When an exercise calls for the 'exact value,' it means that the answer should not be rounded or approximated, but rather the precise answer as a whole number or simplified fraction. An exact value provides the most accurate representation, especially important in algebraic and geometric calculations.
For \(\sqrt{64}\), the exact value is 8, since multiplying 8 by itself gives exactly 64.

• This avoids possible errors due to rounding.
• In computational mathematics, having exact values is crucial for precision.

Understanding when and how to find exact values is an essential skill in mathematics, fostering accuracy in problem-solving.

Multiplication

Multiplication is one of the basic arithmetic operations and a fundamental concept in mathematics. It involves calculating the total of one number added together a specified number of times. For example, if we take the number 8:

• Multiplying 8 by itself means calculating 8 \(\times\) 8.
• This results in the product of 64.

When it comes to square roots, multiplication helps us verify our results. By multiplying 8 by itself and getting 64, we confirm that 8 is indeed the square root of 64. This demonstrates the close relationship between multiplication and the concept of square roots.

Algebraic Concepts

Algebraic concepts form the backbone of many mathematical principles, and comprehending them can enhance your mathematical intuition. Algebra involves using symbols and letters to represent numbers and quantities in equations and expressions.
In the context of evaluating \(\sqrt{64}\), the application of algebra comes into play:

• Understanding that \(\sqrt{x}\) refers to finding a number that, when squared, returns x.
• Using variables to generalize expressions. For example, if \(y = \sqrt{x}\), then \(y^2 = x\).

This approach of breaking down and analyzing expressions and equations helps in solving a wide array of mathematical problems and is a building block for higher-level math like calculus and beyond.