Problem 37
Question
Simplify the expression. \((3 \cdot 7)^{2}\)
Step-by-Step Solution
Verified Answer
Therefore, the simplified form of the expression \((3 \cdot 7)^{2}\) is 441.
1Step 1: Perform operation inside the Parentheses
Let's first carry out the multiplication inside the parentheses: \(3 \cdot 7 = 21\). So, the expression then becomes \(21^{2}\).
2Step 2: Perform the Exponentiation
Next, square of 21 needs to be calculated, which means multiplying 21 by itself: \(21 \times 21 = 441\).
Key Concepts
Order of OperationsExponentiationArithmetic OperationsAlgebraic Skills
Order of Operations
Understanding the order of operations is essential when simplifying algebraic expressions. This set of rules is a fundamental part of mathematics, guiding us on which calculations to perform first to get the correct result.
The basic principle we follow is 'PEMDAS' or 'BODMAS', acronyms that help us remember the sequence: Parentheses or Brackets first, Exponents or Orders next (such as squares and cubes), followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
In the given exercise, the expression \( (3 \cdot 7)^{2} \) demands that we first deal with the operation inside the parentheses (3 times 7), and only then raise the result to the power of 2. Ignoring this order would lead to incorrect results and confuse the true meaning of the expression.
The basic principle we follow is 'PEMDAS' or 'BODMAS', acronyms that help us remember the sequence: Parentheses or Brackets first, Exponents or Orders next (such as squares and cubes), followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
In the given exercise, the expression \( (3 \cdot 7)^{2} \) demands that we first deal with the operation inside the parentheses (3 times 7), and only then raise the result to the power of 2. Ignoring this order would lead to incorrect results and confuse the true meaning of the expression.
Exponentiation
Exponentiation is an operation involving two numbers, the base and the exponent. The base is the number being multiplied by itself, and the exponent tells us how many times the base is used as a factor. For instance, in the expression \( 21^{2} \), 21 is the base and 2 is the exponent.
Hence, exponentiation is merely shorthand for repeated multiplication. Rather than writing \( 21 \cdot 21 \) repetitively, we use \( 21^{2} \) to imply that 21 is multiplied by itself once. Mastering exponentiation is crucial for higher-level mathematics, as it's utilized frequently in algebra, calculus, and scientific notation.
Hence, exponentiation is merely shorthand for repeated multiplication. Rather than writing \( 21 \cdot 21 \) repetitively, we use \( 21^{2} \) to imply that 21 is multiplied by itself once. Mastering exponentiation is crucial for higher-level mathematics, as it's utilized frequently in algebra, calculus, and scientific notation.
Arithmetic Operations
Arithmetic operations are the building blocks of math, consisting of addition, subtraction, multiplication, and division. These four operations are used in various combinations to solve a wide array of mathematical problems.
In our example, we use multiplication within the parentheses to simplify \( 3 \cdot 7 \), and then we square the result, which is another form of multiplication. As we manipulate algebraic expressions, the precision of these basic operations is vital, ensuring that we progress towards the correct solution smoothly and accurately.
In our example, we use multiplication within the parentheses to simplify \( 3 \cdot 7 \), and then we square the result, which is another form of multiplication. As we manipulate algebraic expressions, the precision of these basic operations is vital, ensuring that we progress towards the correct solution smoothly and accurately.
Algebraic Skills
Developing robust algebraic skills is essential for anyone studying mathematics. These skills involve the ability to recognize and apply mathematical properties, manipulate expressions, and solve equations.
One particular skill demonstrated in simplifying the expression \( (3 \cdot 7)^{2} \) is recognizing how to efficiently execute calculations by following the correct order of operations and utilizing exponentiation correctly. Another skill is the ability to convert between different forms of expression, such as turning an exponentiated number into its expanded multiplication form to find the solution. Strong algebraic skills enable students to tackle more complex problems with confidence.
One particular skill demonstrated in simplifying the expression \( (3 \cdot 7)^{2} \) is recognizing how to efficiently execute calculations by following the correct order of operations and utilizing exponentiation correctly. Another skill is the ability to convert between different forms of expression, such as turning an exponentiated number into its expanded multiplication form to find the solution. Strong algebraic skills enable students to tackle more complex problems with confidence.
Other exercises in this chapter
Problem 37
Write the number in scientific notation. the number $$ 88,000,000 $$
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Graph the exponential function. $$y=-3(8)^{x}$$
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Evaluate the expression without using a calculator. $$ (1 \cdot 7)^{-3} $$
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In Exercises 38–41, use the following information. From 1894 to 1903 the number of miles of cable car track in the United States decreased by about 11% per year
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