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Chapter 6: Problem 1

Evaluate the expressions without using a calculator. $$ 3 \cdot 2^{3} $$

Short Answer

Expert verified
Question: Evaluate the expression $$3 \cdot 2^{3}$$ without using a calculator. Answer: 24

Step by step solution

01

Evaluate the exponent

In this expression, we have an exponent \(2^3\). We need to calculate the value of this exponent first before proceeding with the multiplication. To find the value of \(2^3\), multiply the base (2) by itself for the number of times of the exponent (3). So, $$2^3 = 2 \cdot 2 \cdot 2 = 8$$.
02

Multiply the exponent's value with the constant

Now that we know the value of the exponent (\(2^3 = 8\)), we can proceed to multiply it with the given constant (3). $$3 \cdot 2^{3} = 3 \cdot 8$$.
03

Simplify the expression

Finally, we need to multiply 3 by 8 to get the simplified expression: $$3 \cdot 8 = 24$$. The expression $$3 \cdot 2^{3}$$ evaluates to 24 without using a calculator.

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Key Concepts

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

Order of Operations
When solving math problems, it is important to follow the rules of **order of operations** to get the correct answer. These rules tell us in which order to solve different parts of an expression. Understanding and applying these rules helps us simplify complex problems accurately. The order of operations is commonly remembered by the acronym **PEMDAS**, which stands for:
  • Parentheses: First, solve anything inside parentheses or brackets.
  • Exponents: Next, handle exponents or powers.
  • Multiplication and Division: After exponents, do multiplication and division from left to right.
  • Addition and Subtraction: Lastly, carry out addition and subtraction from left to right.
In the exercise, the expression was \(3 \cdot 2^3\). According to the order of operations, we first solve the exponent \(2^3\) before the multiplication.
Simplifying Expressions
**Simplifying expressions** involves reducing them to their simplest form without changing their value. This process ensures that the expression is easier to understand and solve.
  • Identify any operations within the expression that need to be addressed, such as exponents or basic arithmetic.
  • Solve these operations step-by-step, always keeping the order of operations in mind.
  • Continue combining terms until the expression is fully simplified.
In our exercise, after calculating \(2^3 = 8\), we simplified the expression by performing the multiplication \(3 \cdot 8\), arriving at the simple result of 24. This shows how simplifying serves to make calculations clearer and more manageable.
Basic Arithmetic
**Basic arithmetic** is the foundation of all mathematics, dealing with fundamental operations such as addition, subtraction, multiplication, and division. Mastery of these operations is crucial for evaluating expressions and solving problems efficiently. Here's a simple overview of these operations:
  • Addition combines numbers to obtain their total.
  • Subtraction finds the difference between numbers.
  • Multiplication involves repeated addition of the same number.
  • Division is the process of distributing a number into equal parts.
In the given exercise, we focused on multiplication as part of our calculation procedure. After determining that \(2^3 = 8\), we simply multiplied it by 3 using basic multiplication to arrive at the final solution of 24. Understanding basic arithmetic ensures that no step is accidentally skipped or performed incorrectly.

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Most popular questions from this chapter

Rewrite each expression by rationalizing the denominator. $$ \frac{\sqrt{3}}{3 \sqrt{2}+\sqrt{3}} $$

Write with a single exponent. $$ \frac{a^{x}}{b^{x}} $$

Write each expression as a power raised to a power. There may be more than one correct answer. $$ 5^{x^{2}} $$

Combine radicals, if possible. \(-6 \sqrt{98}+4 \sqrt{8}\)

In Exercises \(11-16,\) write the expression as an equivalent expression in the form \(x^{n}\) and give the value for \(n\). $$ \frac{1}{\sqrt{x}} $$
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