Problem 29 - Algebra 1

Question

Evaluate the power. $$ 6^{4} $$

Step-by-Step Solution

Answer

1296

Step 1: Understanding Power

A power is an expression that tells us to multiply a number by itself a certain amount of times. Here, the base is 6, and the exponent is 4, meaning we have to multiply 6 by itself 4 times.

Step 2: Perform the Calculations

Now, perform the successive multiplication: \(6 \times 6 = 36\), then \(36 \times 6 = 216\) and finally \(216 \times 6 = 1296\)

Step 3: Writing down the Final Answer

According to the calculations we performed, the fourth power of 6 is 1296

Key Concepts

Powers in Algebra

In algebra, powers are a fundamental concept used to represent repeated multiplication of the same number. When we talk about powers, we refer to expressions like \( 6^4 \), where '6' is the base and '4' is the exponent. The power tells us to multiply the base by itself as many times as indicated by the exponent.

For instance, \( 6^4 \) equals 6 multiplied by itself four times: \( 6 \times 6 \times 6 \times 6 \). Understanding this notation is essential for tackling a wide range of algebraic problems, from simple calculations to more complex equations involving exponents.

Evaluating Exponents

Evaluating exponents involves following a specific set of rules to find the value of an exponential expression. The exponent indicates the times a number, which is the base, is multiplied by itself. In the example of \( 6^4 \), we evaluate by multiplying 6 four times successively.

The process of evaluation can also involve dealing with negative exponents, which imply division, or fractional exponents, which correspond to roots. Remember that any number raised to the zeroth power is 1, and any non-zero number raised to the first power is the number itself.

Successive Multiplication

Successive multiplication is the process used when we evaluate powers. It is a systematic way of multiplying the base number repeatedly. For example, when calculating \( 6^4 \), you start with the base 6 and multiply it by itself: \( 6 \times 6 \), to get 36. Now use that result and multiply by the base again: \( 36 \times 6 \), to get 216. One more multiplication, \( 216 \times 6 \), gives us the final result of 1296.

This methodical approach ensures that each step is simple multiplication, and we achieve the accurate final result without skipping any multiplication.

Exponential Expressions

Exponential expressions, like \( 6^4 \), encode a lot of information in a compact notation used widely in algebra. It involves a base, in this case, 6, and an exponent, here 4. These expressions are not limited to numbers; we also use variables and can see expressions like \( x^4 \) or \( y^8 \).

Understanding how to work with exponential expressions is crucial for algebraic operations such as expansion, factoring, and simplifying. They are also the basis for more advanced topics, including exponential growth and decay in areas such as finance and science.