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Chapter 1: Problem 66

Simplify the given expression. $$ 6 \cdot 3^{2}+7 \cdot 5+12 $$

Short Answer

Expert verified
The simplified expression is 101.

Step by step solution

01

Calculate the Powers

Identify and calculate the powers in the expression. The first power is \(3^2\). Calculate: \(3^2 = 3 \times 3 = 9\). Substitute it back into the expression to get: \[6 \cdot 9 + 7 \cdot 5 + 12\]
02

Multiply Through the Expressions

Now, perform the multiplications. Start with the first part: Calculate \(6 \cdot 9 = 54\).Substitute the result into the expression:\[54 + 7 \cdot 5 + 12\]Next, compute \(7 \cdot 5 = 35\).Substitute back into the expression:\[54 + 35 + 12\]
03

Add the Results

Now, add up the simplified components in the expression.First, add \(54 + 35 = 89\).Substitute the result back into the expression:\[89 + 12\]Finally, compute \(89 + 12 = 101\).

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Key Concepts

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

Order of Operations
When simplifying expressions, it鈥檚 crucial to follow the order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)). Incorrectly applying the order can lead to wrong answers. Following PEMDAS ensures that you perform calculations in the correct sequence, which is essential for accurate simplification.
In the exercise, the expression \(6 \cdot 3^{2} + 7 \cdot 5 + 12\) was simplified using the order of operations:
  • First, the exponent \(3^{2}\) was calculated.
  • Next, multiplication was performed.
  • Finally, the numbers were added together.
This sequence kept the calculations organized and accurate. Be sure to practice the order to simplify expressions efficiently.
Powers and Exponents
Powers and exponents are an essential part of many mathematical operations. A power consists of a base and an exponent. The base is the number being multiplied by itself, and the exponent tells you how many times to multiply it. For example, in \(3^{2}\), the base is 3, and the exponent is 2.
Understanding and correctly calculating powers is crucial. In the original exercise, \(3^{2}\) was the first step in simplification. This involved multiplying the base, 3, by itself to get 9:
  • Calculate \(3 \times 3\)
  • Result is 9, which replaces \(3^{2}\) in the expression.
Mastering exponents will not only help you with specific exercises but also lay a strong foundation for more advanced math topics.
Arithmetic Operations
Arithmetic operations, including addition, subtraction, multiplication, and division, form the core of mathematical calculations. Knowing how to perform these operations correctly is essential for simplifying expressions and solving problems.
In the step-by-step solution, various arithmetic operations were applied:
  • Multiplication: Once the exponent was solved, multiplication followed. For example, \(6 \cdot 9\) = 54, and \(7 \cdot 5 = 35\).
  • Addition: After the multiplications, the results were added: \(54 + 35\), and then the result 89 was added to 12, making 101.
Practicing these skills ensures accuracy in calculations and aids in understanding more complex mathematical concepts.

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