Problem 10
Question
Evaluate the expression. $$ 2^{3} \cdot 3^{2} $$
Step-by-Step Solution
Verified Answer
The evaluation of the given expression is 72.
1Step 1: Solve \(2^{3}\)
Firstly, calculate the value of the exponent \(2^{3}\). This represents \(2 \cdot 2 \cdot 2\). Therefore, \(2^{3} = 8\).
2Step 2: Solve \(3^{2}\)
Next, calculate the value of the exponent \(3^{2}\). This represents \(3 \cdot 3\). Therefore, \(3^{2} = 9\).
3Step 3: Multiply the Results
Now, multiply the results from the first two steps together. \(8 \cdot 9 = 72\).
Key Concepts
ExponentiationOrder of OperationsMultiplying Integers
Exponentiation
Exponentiation is a mathematical operation, involving two numbers, the base and the exponent. When you encounter an expression such as \(2^{3}\), the number 2 is the base and 3 is the exponent, and it instructs you to multiply 2 by itself 3 times. Therefore, \(2^{3} = 2 \times 2 \times 2 = 8\). Similarly, for \(3^{2}\), 3 is the base and 2 is the exponent, showing that 3 should be multiplied by itself, resulting in \(3^{2} = 3 \times 3 = 9\).
Understanding exponentiation is crucial because it simplifies expressions and lays the groundwork for more advanced mathematical concepts like polynomial multiplication and powers of functions.
Understanding exponentiation is crucial because it simplifies expressions and lays the groundwork for more advanced mathematical concepts like polynomial multiplication and powers of functions.
Order of Operations
The order of operations is a set of rules that dictate the sequence in which the parts of a mathematical expression should be evaluated. The standard convention follows the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In the expression \(2^{3} \cdot 3^{2}\), you must first evaluate the exponents before proceeding to multiplication. By following the correct order, you ensure the accurate calculation of expressions and avoid common mistakes that can change the result drastically. Always remember to apply the order of operations when solving math problems to get to the correct answer.
In the expression \(2^{3} \cdot 3^{2}\), you must first evaluate the exponents before proceeding to multiplication. By following the correct order, you ensure the accurate calculation of expressions and avoid common mistakes that can change the result drastically. Always remember to apply the order of operations when solving math problems to get to the correct answer.
Multiplying Integers
Multiplying integers might seem straightforward, but it is essential to understand the rules that apply. When you multiply two positive numbers, the result is positive. In the given problem, after calculating the exponents, you multiply the results: \(8 \cdot 9 = 72\).
Remember that these rules also extend to negatives — when you multiply a positive number by a negative, the result is negative, and when you multiply two negative numbers, the result is positive. Keeping these rules in mind is fundamental as you progress through algebraic expressions and equations involving multiplication of integers.
Remember that these rules also extend to negatives — when you multiply a positive number by a negative, the result is negative, and when you multiply two negative numbers, the result is positive. Keeping these rules in mind is fundamental as you progress through algebraic expressions and equations involving multiplication of integers.
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