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Chapter 10: Problem 29

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$(\sqrt{33})^{2}$$

Short Answer

Expert verified
33

Step by step solution

01

- Understand the Expression

The given expression is \((\sqrt{33})^{2}\). Our goal is to simplify or evaluate this expression.
02

- Apply the Property of Squares and Square Roots

Recall that \(\left(\sqrt{a}\right)^{2} = a\). This property states that the square of the square root of a number is the number itself.
03

- Simplify the Expression

By applying the property, the expression \((\sqrt{33})^{2}\) simplifies to 33.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
When dealing with square roots, remember that a square root of a number is a value that, when multiplied by itself, gives the original number. The square root is denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{}36 = 6 \), because \( 6 \times 6 = 36 \). Understanding square roots is essential as it allows simplifying expressions containing square roots in a step-by-step manner.

Here are key points to remember about square roots:
  • Every non-negative number has a non-negative square root.
  • Square roots and squares are inverse operations.
  • If you know how to square a number, you can find its square root, and vice versa.
If we look at our exercise \( \sqrt{}33 \), we observe it is the principal (non-negative) square root of 33. This concept is fundamental in algebra and helps in solving problems involving square roots.
Properties of Exponents
Exponents are used to represent repeated multiplication of the same number. In mathematical notation, you will often see expressions like \( x^n \), which means multiplying x by itself n times. Exponents follow certain properties that make evaluating expressions straightforward.

Here are some essential properties of exponents to keep in mind:
  • Product of Powers: \( a^m \times a^n = a^{m+n} \)
  • Power of a Power: \( (a^m)^n = a^{mn} \)
  • Power of a Product: \( (ab)^n = a^n \times b^n \)
  • Zero Exponent: \( a^0 = 1 \) (where a 鈮� 0)
For the given exercise \( (\sqrt{}33)^2 \), we apply the Power of a Power property. Since the exponent and the radical (square root) are inverse functions, they cancel each other out, leading to the simplified result. This makes it easier to handle expressions involving both radicals and exponents.
Algebraic Evaluation
Simplifying algebraic expressions involves reducing them to their most basic form. This often includes combining like terms, using properties of exponents, and performing basic arithmetic operations. The goal is to make the expression as simple as possible.

Here are steps to follow for algebraic evaluation:
  • First, look for terms that can be combined.
  • Next, apply the properties of exponents, as these often help to reduce the complexity.
  • Finally, carry out the arithmetic to get the simplest form of the expression.
Analyzing our specific exercise, \( (\sqrt{}33)^2 \), we apply the principle of powers and roots being inverse operations. By knowing that \( \sqrt{}x \times \sqrt{}x = x \), we simplify the expression immediately to 33. This process illustrates how understanding the properties and systematically applying them leads to the correct and simplified result.

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