Problem 28 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (\sqrt{100})^{3} $$

Step-by-Step Solution

Answer

The result of the expression \( (\sqrt{100})^{3} \) is 1000.

Step 1: Evaluate the Square Root

Firstly, evaluate the square root part of the expression which is \( \sqrt{100} \). The square root of 100 is 10 since 10*10 equals 100.

Step 2: Apply the Exponential

After determining the square root, the next step is to raise the result to the power of 3. Hence, calculate \( 10^{3} \), which simplifies to 10*10*10.

Step 3: Solve the expression

To get the final answer, calculate the value of \(10*10*10\), which equals 1000.

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics. They allow us to reverse the process of squaring a number. For example, the square root of 100, written as \( \sqrt{100} \), is 10.
This is because if you multiply 10 by itself (\(10 \times 10\)), you get back the original number, which is 100.
To find a square root, look for a number that multiplied by itself equals the desired number inside the square root sign. Square roots are often denoted by the radical symbol \( \sqrt{} \).

• \( \sqrt{16} = 4 \) (because \(4 \times 4 = 16\))
• \( \sqrt{25} = 5 \) (because \(5 \times 5 = 25\))

Knowing how to evaluate square roots is necessary for tackling more complex mathematical expressions.

Basics of Arithmetic

Arithmetic is the branch of mathematics that involves basic operations like addition, subtraction, multiplication, and division. It serves as the foundation for more advanced topics like algebra and calculus. In this exercise, arithmetic is used when calculating \(10 \times 10 \times 10\) after evaluating the square root. To strengthen your arithmetic skills:

• Practice the order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Work on mental math to improve speed and accuracy in performing arithmetic operations.Arithmetic can help solve a wide range of everyday problems, from budgeting to cooking.

With arithmetic, solving expressions becomes a walk in the park.

Evaluating Mathematical Expressions

Mathematical expression evaluation involves simplifying expressions using a set of rules and operations to find their values. This process includes interpreting symbols, calculating values, and understanding the hierarchy of operations.In the given expression, \((\sqrt{100})^3\), the steps to evaluate are:

• First, find the square root \( \sqrt{100} \rightarrow 10 \)
• Then, raise the result to the power of 3 \(10^3\) Raising a number to a power means multiplying it by itself the number of times indicated by the exponent.
• Finally, solve \(10 \times 10 \times 10 = 1000\)

Evaluating expressions requires careful attention to each step, ensuring all operations are performed correctly and in the right order.Being precise in mathematical evaluation helps in achieving accurate results and in understanding mathematical concepts thoroughly.