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Chapter 2: Problem 25

For Problems \(21-40\), simplify each numerical expression. $$ 5(2)^{2}-4(2)-1 $$

Short Answer

Expert verified
The simplified expression is 11.

Step by step solution

01

Evaluate the Powers

First, simplify the power in the expression. Calculate \(2^2\) which equals \(4\). So the expression becomes \(5(4) - 4(2) - 1\).
02

Multiply

Now multiply the numbers. First, multiply \(5\) by \(4\), which equals \(20\). Next, multiply \(4\) by \(2\), which equals \(8\). The expression now looks like this: \(20 - 8 - 1\).
03

Subtract

Subtract the numbers in the order they appear. First, subtract \(8\) from \(20\) to get \(12\). Then, subtract \(1\) from \(12\) to get \(11\).

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Key Concepts

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

Order Of Operations
Simplifying expressions can be fun and easy once you know the rules. The order of operations is a crucial concept to understand when simplifying numerical expressions. This rule provides a consistent way to evaluate an expression that involves more than one operation.
Here's the order you should follow:
  • Parentheses (do any calculations inside them first)
  • Exponents (evaluate powers and square roots)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
Using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) will help you remember this order. In the expression given for the exercise, you start with the exponentiation, then move to multiplication, and finally subtraction鈥攆ollowing this order ensures you simplify it correctly.
Powers and Exponents
Understanding exponents is key to solving numerical expressions effortlessly. In mathematics, a power or exponent indicates how many times a number (the base) is multiplied by itself. For example, in the expression \(2^2\), the exponent is \(2\), and the base is \(2\). Here, since \(2^2\) means \(2\times2\), the result is \(4\).
Let鈥檚 break this down:
  • The base is the number being multiplied.
  • The exponent tells you how many times to multiply the base by itself.
  • If you have \(2^3\), it means \(2\times2\times2\), which equals \(8\).
In the exercise, evaluating the power \(2^2\) was the first step. It became \(4\), simplifying the expression to \(5(4) - 4(2) - 1\). Mastering exponents builds a foundation for more complex algebraic expressions.
Subtraction in Arithmetic
Subtraction is one of the four elementary operations in arithmetic. It is the process of taking one number away from another. In a sequence of operations, subtraction is performed after multiplication and division unless it's in parentheses.
Here's how you approach subtraction:
  • Start with the first number (the minuend).
  • Subtract the second number (the subtrahend) from the minuend.
  • Continue subtracting any remaining numbers in sequence.
In the exercise, after multiplying, we simplify the expression to a sequence of subtractions: \(20-8-1\). First, \(20 - 8\) is \(12\), and then subtracting \(1\) gives \(11\), which completes the solution. Understanding and practicing these calculations is crucial to mastering basic arithmetic.

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Most popular questions from this chapter

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