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Chapter 1: Problem 58

Simplify the given expression. $$ (5-3)^{4} $$

Short Answer

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16

Step by step solution

01

Simplify Inside the Parentheses

First, we need to evaluate the expression inside the parentheses. The expression is \(5-3\). Subtract \(3\) from \(5\) to get \(2\).
02

Apply the Exponent

Now that we have simplified the expression inside the parentheses, we have \(2^4\). We need to calculate \(2\) raised to the power of \(4\). This means we multiply \(2\) by itself \(4\) times: \(2 \times 2 \times 2 \times 2\).
03

Perform Multiplication

Multiply the numbers: \(2 \times 2 = 4\), then \(4 \times 2 = 8\), and finally \(8 \times 2 = 16\). Thus, \(2^4 = 16\).

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Key Concepts

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

Parentheses
Dealing with parentheses in math is like opening a box to find the surprise inside. It's where your operation journey begins. Parentheses are used to group parts of an expression that need to be solved first. Think of them as priority markers in a mathematical sentence.
  • Always tackle operations inside parentheses first before anything else.
  • This is according to the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In our example, the expression inside the parentheses is \(5 - 3\). We simplify this to \(2\) before applying other operations. It's like tidying your room before having the space to play!
Simplification
Simplification is the art of making expressions easier to understand and solve. It involves breaking down complex parts into simpler ones, so they are more manageable. In mathematics, this often means performing operations like addition or subtraction inside of parentheses first, then moving on to exponents or other operations.
  • This step reduces the problem to its simplest form.
  • In equations involving exponents, the simplified form is crucial for further calculations.
By reducing \((5 - 3)^4\) to \(2^4\), we made the problem straightforward and ready for exponentiation. Simplification helps us see the path clearly and is a key skill in solving all kinds of math problems!
Multiplication
Multiplication is one of the fundamental operations in arithmetic. When dealing with exponents, multiplication takes center stage, especially after simplifying terms inside the parentheses. The key to multiplication with exponents is understanding what the exponent signifies:
  • It tells you how many times to multiply the base by itself.
  • For our problem \(2^4\), this means multiplying \(2\) by itself four times: \(2 \times 2 \times 2 \times 2\).
One step at a time, multiplication builds to find the answer: first \(2 \times 2 = 4\), then \(4 \times 2 = 8\), and finally \(8 \times 2 = 16\). This step culminates in the answer to the expression, showing how multiplication combined with exponents simplifies our mathematical life! It's a repeating action that derives power to bring us to the final solution.

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