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Chapter 2: Problem 31

Simplify. See Examples 1 through \(10\). $$ 7 \cdot 6^{2}+4 $$

Short Answer

Expert verified
The simplified expression is 256.

Step by step solution

01

Evaluate the exponentiation

Start by solving the exponentiation part of the expression. The expression given is \(6^2\), which means 6 raised to the power of 2. Calculate \(6^2 = 6 \times 6 = 36\).
02

Multiply with the coefficient

Now that we have \(6^2 = 36\), substitute back into the original expression to get \(7 \cdot 36 + 4\). Next, multiply 7 by 36 to get \(7 \times 36 = 252\).
03

Add the remaining integer

After completing the multiplication, you have \(252\) from the previous step. Now, add the remaining integer, which is 4, to it. Calculate \(252 + 4 = 256\).

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Key Concepts

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

Exponentiation
Exponentiation involves raising a number, known as the base, to the power of another number, the exponent. It is a shorthand way of expressing repeated multiplication. For example, when you see an expression like \(6^2\), it signifies that the base 6 should be multiplied by itself two times. Thus, \(6^2 = 6 \times 6 = 36\). This operation serves as an important foundation for simplifying mathematical expressions as it focuses on growing numbers rapidly by powers rather than through simple repeated addition only.
  • Base and Exponent: In \(a^b\), 'a' is the base and 'b' is the exponent.
  • Common Exponents: Any number raised to the power of 2 is called "squared," and to the power of 3 is "cubed."
To solve an exponentiation problem, calculate what the base number becomes when it is multiplied by itself as many times as indicated by the exponent.
Multiplication
Once you have solved any exponentiation in your expression, multiplication is the next step. Multiplication is one of the basic arithmetic operations and is essentially repeated addition. In our example, after determining that \(6^2 = 36\), the expression becomes \(7 \cdot 36\). This means we need to add 36 to itself a total of seven times:
  • Think of multiplication as adding the same number repeatedly.
  • Steps: To multiply 7 and 36, calculate the total as \(7 \times 36\ = 252\).
Multiplication simplifies complex addition operations efficiently, especially when dealing with larger numbers or multiple steps. It is an essential element in simplifying mathematical expressions together with other operations.
Addition
After handling both exponentiation and multiplication, addition is often the final step in simplifying expressions. Addition is the process of finding the total or sum by combining numbers. In this context, once the product of the multiplication step is determined, it is simply about adding any remaining terms.
  • Order of Operations: Addition usually follows multiplication and exponentiation in expressions.
  • For our expression \(7 \cdot 6^2 + 4\), after reaching \(252\) from multiplication, the next step is simply \(252 + 4\), resulting in a total of \(256\).
Here's an important point: Always remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), to ensure proper simplification of expressions.

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Most popular questions from this chapter

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