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Chapter 2: Problem 52

Simplify. See Examples 1 through \(10\). $$ \left(11-3^{2}\right)^{3} $$

Short Answer

Expert verified
The simplified value is 8.

Step by step solution

01

Evaluate the Exponentiation Inside the Parentheses

First, calculate the exponentiation inside the parentheses. We have the expression \(3^2\). Calculate \(3^2 = 3 \times 3 = 9\).
02

Perform Operations Inside the Parentheses

Now, substitute \(3^2\) with \(9\) in the original expression inside the parentheses: \(11 - 3^2 = 11 - 9\). Calculate \(11 - 9 = 2\).
03

Raise the Result to the Third Power

After solving the expression inside the parentheses (which is \(2\)), raise it to the power of 3: \(2^3\). Calculate \(2^3 = 2 \times 2 \times 2 = 8\).

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Key Concepts

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

Understanding the Order of Operations
When solving mathematical expressions, it's vital to follow the **order of operations**. This ensures that everyone solves the problem the same way and gets the same result. The order of operations is often remembered by the acronym PEMDAS:
  • **P**arentheses
  • **E**xponents (or powers)
  • **M**ultiplication and **D**ivision (from left to right)
  • **A**ddition and **S**ubtraction (from left to right)
In the given expression, \((11-3^{2})^{3}\), we need to first work inside the parentheses. **Exponents** within the parentheses must be calculated first, ensuring we follow PEMDAS. Once the inside is simplified, we can then move onto handling the exponent outside the parentheses. This sequence assures that we systematically simplify the expression, leading to an accurate final result.
Simplifying Expressions Step by Step
**Simplifying** mathematical expressions makes them easier to understand. It involves writing an expression in its simplest form by performing operations inside any parentheses first, resolving any exponents, and then carrying out multiplication or division, followed by addition or subtraction. For the exercise expression: \((11-3^{2})^{3}\), the process is all about simplifying stepwise:
  • First, tackle the exponent inside the parentheses: calculate \(3^{2} = 9\).
  • Then, subtract inside the parentheses: \(11 - 9 = 2\).
  • Finally, apply the exponent outside the parentheses by calculating \(2^{3}\) as the last simplifying step.
These steps allow us to turn complicated problems into manageable parts, ultimately finding the solutions more swiftly and accurately.
Exploring Powers of Numbers
In mathematics, a **power** represents how many times a number, called the base, is multiplied by itself. The notation \(a^b\) is read as "a raised to the power of b", where "a" is the base and "b" is the exponent. For example, in the term \(3^{2}\), 3 is the base and 2 is the exponent. We multiply 3 by itself to get 9. Hence, \(3^{2} = 9\). Exploring how powers work is crucial for deepening mathematical understanding:
  • A power of 2 is often called "squared" because it represents a two-dimensional square layout.
  • A power of 3 is known as "cubed" as it illustrates a three-dimensional cube shape.
  • Using powers simplifies repeated multiplication, making calculations neater and more efficient.
By understanding powers, not only do calculations become simpler, but you also develop a foundational understanding needed for advanced math topics.

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