Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents.. $$ \sqrt{81} $$

Step-by-Step Solution

Answer

The simplified expression without radicals or exponents is \( 9 \).

Step 1: Identify the square root of 81

The square root of 81 is a number that, when multiplied by itself, equals 81. In this case, it is 9 because \( 9 * 9 = 81 \).

Step 2: Rewrite the expression without radicals or exponents

Since the square root of 81 is 9, we can rewrite the expression \( \sqrt{81} \) as 9. So, the simplified expression without radicals or exponents is: \( 9 \)

Key Concepts

Square Roots

Square roots are mathematical functions symbolized by the radical sign \( \sqrt{ } \). They represent a value that, when multiplied by itself, gives the original number. For example, the square root of 81 is 9 because \( 9 \times 9 = 81 \).
Square roots are fundamental in solving equations and simplifying mathematical expressions. They are often used to "undo" the process of squaring a number.

• This means if you have a number squared, taking the square root returns the number to its original form.
• Mathematically, it is expressed as \( x^2 \rightarrow x \) when \( \sqrt{ } \) is applied.

Knowing how to compute square roots is crucial in many areas, including geometry and algebra. There are various methods to find square roots, such as estimation, the use of calculators, or memorizing common square roots like \( \sqrt{81} = 9 \).

Mathematical Expressions

A mathematical expression is a combination of numbers, operators, symbols, and sometimes variables, all arranged in a meaningful way to represent a mathematical concept or problem.
Expressions can vary in complexity, from simple numerical identities like \( 4 + 3 \) to complex algebraic equations.
In the context of simplifying radicals, expressions often contain square roots or exponents.

• Simplifying these expressions involves using mathematical rules and operations to break them down into simpler forms.
• For example, converting \( \sqrt{81} \) to 9 is a simplification process that eliminates the radical symbol.

Understanding how to manipulate and rewrite expressions is essential for solving equations and completing algebraic operations.

Number Operations

Number operations are the building blocks of arithmetic and involve addition, subtraction, multiplication, and division. They play a critical role in manipulating numbers and solving problems.
Each operation has its unique properties and rules that govern how numbers are combined and transformed.

• For instance, multiplication is repeated addition, and division is the inverse of multiplication.
• Understanding how to perform these operations seamlessly is crucial for simplifying radicals and solving equations.

When dealing with square roots like \( \sqrt{81} \), multiplication comes into play because the square root itself represents repeated multiplication: \( 9 \times 9 = 81 \).
This knowledge assists in recognizing patterns and manipulating expressions efficiently. Mastery of number operations is foundational to further mathematical learning and problem-solving.