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Chapter 5: Problem 53

Evaluate. \(3 \cdot 4^{2}\)

Short Answer

Expert verified
The value is 48.

Step by step solution

01

Understand the Expression

The expression we have to evaluate is \(3 \cdot 4^2\). This means we need to multiply 3 by the square of 4.
02

Calculate the Exponent

The exponent part of our expression is \(4^2\). We need to calculate the square of 4, which is \(4 \times 4 = 16\).
03

Multiply the Results

Now, take the result from Step 2, which is 16, and multiply it by 3. So, \(3 \times 16 = 48\).
04

Conclusion

After evaluating all parts of the expression, we find that the value of \(3 \cdot 4^2\) is 48.

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics. They are used to express repeated multiplication of a number by itself. In the expression, the exponent is the small number located at the upper right corner of the base number. For example, in \(4^2\), the number 4 is the base and 2 is the exponent. This means you multiply 4 by itself:
  • \(4^2\) = \(4 \times 4\)
  • The result is 16
Exponents make it easier to write and work with large numbers. Think of exponents as shorthand for saying "this number is multiplied by itself so many times."
In our exercise, the exponent 2 tells us to multiply 4 by itself, leading to the value 16.
Multiplication
Multiplication is one of the basic arithmetic operations. It is essentially a shortcut for addition. For instance, instead of adding 4 four times (like in the expression \(4 + 4 + 4 + 4\)), you multiply 4 times 4.
  • Multiplication is denoted by symbols such as \(\times\) or \(\cdot\)
  • In the exercise, \(3 \cdot 16\) signifies multiplying 3 by 16
The result of multiplication is called the product. It provides a faster way to add numbers that are the same.
For our task, after calculating the exponent to get 16, we multiplied it by 3 to get the final result of 48.
Order of Operations
The order of operations is a set of rules that indicates the sequence in which mathematical operations should be performed. This is crucial to solving expressions correctly. The standard rule is remembered by the acronym PEMDAS:
  • Parentheses first
  • Exponents (i.e., powers and roots, etc.)
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)
For the given problem \(3 \cdot 4^2\), according to PEMDAS, you first calculate the exponent \(4^2\), as it precedes multiplication in the order of operations. After solving the exponent, you proceed to multiplication \(3 \cdot 16\).
This approach ensures that everyone arrives at the same final answer of 48, maintaining consistency in mathematical calculations.

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Most popular questions from this chapter

Multiply. $$(a+4)\left(a^{2}-6 a+6\right)$$

Add or subtract as indicated. See Examples 9 through \(13 .\) $$ \left(x^{2}+2 x y-y^{2}\right)+\left(5 x^{2}-4 x y+20 y^{2}\right) $$

Multiply each expression. $$ -4 a\left(3 a^{2}-4\right) $$

Multiply. See Section 5.1 $$ -z^{2} y(11 z y) $$

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \frac{1.4 \times 10^{-2}}{7 \times 10^{-8}} $$
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