Problem 69 - Algebra A Combined Function

Question

Evaluate each expression. $$ 3^{4} $$

Step-by-Step Solution

Answer

The value of \(3^4\) is 81.

Step 1: Understand the Base and Exponent

The expression \(3^4\) is composed of a base 3 and an exponent 4. The exponent indicates that the base is multiplied by itself a specific number of times.

Step 2: Expand the Exponentiation

To solve \(3^4\), we need to multiply the base 3 by itself 4 times. This means \(3^4 = 3 \times 3 \times 3 \times 3\).

Step 3: Perform Sequential Multiplication

Start multiplying sequentially: first calculate \(3 \times 3 = 9\), then multiply the result by the next 3: \(9 \times 3 = 27\). Finally, multiply this result by the last 3: \(27 \times 3 = 81\).

Step 4: Confirm the Result

Verify that all steps have been followed correctly and confirm that the final result matches the expected calculation: \(3 \times 3 = 9\), \(9 \times 3 = 27\), and \(27 \times 3 = 81\).

Key Concepts

Understanding Base and Exponent

In exponentiation, the expression is composed of two primary parts: the base and the exponent. The base is the number that is subject to repeated multiplication. The exponent indicates how many times the base is multiplied by itself. For instance, in the expression \(3^4\), the number 3 is the base, and the number 4 is the exponent. This means that we are to multiply 3 by itself 4 times. Exponents simplify expressions of repeated multiplication by summarizing them in a compact form.

The Role of Multiplication in Exponentiation

Multiplication is a mathematical operation where a number, known as the multiplicand, is added to itself a number of times as defined by another number, the multiplier. In exponentiation, multiplication is at the core because the base is multiplied by itself as many times as the number of the exponent. Taking our example \(3^4\), multiplication occurs as follows: \(3 \times 3\), then that product is multiplied by another 3, and so forth. Hence, multiplication turns a lengthy addition process into a more efficient evaluation.

Performing Sequential Multiplication

Sequential multiplication involves performing multiplication step-by-step, one pair of numbers at a time. In the example of \(3^4\), we start by multiplying the first two 3's: \(3 \times 3 = 9\). Then, take this result (9) and multiply by the next 3: \(9 \times 3 = 27\). Finally, multiply 27 by the last 3 to yield the final answer: \(27 \times 3 = 81\). By breaking it into sequential steps, we simplify the process and reduce errors, ensuring accurate results.