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Chapter 1: Problem 2

Evaluate each expression without using a calculator. $$ \left(5^{2} \cdot 4\right)^{2} $$

Short Answer

Expert verified
10,000

Step by step solution

01

Evaluate the Inner Parentheses

First, solve the expression inside the parentheses, which is \(5^{2} \cdot 4\). Calculate \(5^{2}\) first: \(5^{2} = 25\). Then multiply by 4: \(25 \cdot 4 = 100\). So, \((5^{2} \cdot 4) = 100\).
02

Evaluate the Square of the Result

Now take the result from the inner parentheses, which is 100, and evaluate its square. Calculate \(100^{2}\) which equals \(100 \times 100 = 10,000\).

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Key Concepts

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

Order of Operations
When solving mathematical expressions, the order of operations is crucial to arriving at the correct answer. A common way to remember this order is the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Each step is performed from left to right.
  • Start with **Parentheses** - Solve expressions inside parentheses first. This ensures that all operations within the grouping symbols are completed before anything else.
  • Next, handle **Exponents** such as squared numbers. Calculating these early in the process helps reduce complex expressions.
  • Follow with **Multiplication** and **Division** from left to right. They are performed in order as they appear in the expression.
  • Finally, carry out **Addition** and **Subtraction** from left to right, also maintaining the sequence from the expression.
It鈥檚 important to follow this specific order to achieve the right result, as demonstrated in the exercise where the contents of the parentheses were evaluated before applying the squared exponent.
Multiplication
Multiplication is a fundamental arithmetic operation used to calculate repeated additions quickly. In the provided exercise, multiplication is employed to combine terms within the parentheses.
  • Understanding **Basic Multiplication** - To multiply numbers, you need to calculate the total of one number added to itself a specific number of times. For example, in the exercise, multiplying 25 by 4 involves adding 25 four times: 25 + 25 + 25 + 25.
  • **Order Matters** - As seen in the problem, multiplication occurs after calculating exponents but before applying additional exponents to the result. This order ensures mathematical precision.
Mastering multiplication, especially as it applies within parentheses, is key to solving more complex expressions correctly.
Squared Numbers
Squared numbers are the result when a number is multiplied by itself. This operation is indicated by a small "2" positioned just above and to the right of the number, known as an exponent.
  • Understanding **Exponents** - In the exercise, 5 is squared, meaning 5 is multiplied by itself: 5 脳 5 = 25. It's an efficient way of expressing multiplication of a number by itself.
  • **Further Application** - After determining the expression inside the parentheses and evaluating its value as 100, squaring 100 results in multiplying 100 by itself to get 10,000.
Using squared numbers simplifies and accelerates complex calculations, vital for solving equations accurately, as demonstrated in the step-to-step solution of the exercise.

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Most popular questions from this chapter

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