Problem 27 - Algebra 1

Question

Evaluate the power. $$ 5^{2} $$

Step-by-Step Solution

Answer

The value of \(5^{2}\) is 25.

Step 1: Identify the Base and Exponent

The base is the number that is being multiplied and the exponent tells us how many times to multiply the base. In this case, the base is 5 and the exponent is 2.

Step 2: Apply the Power Rule

The power rule states that in order to evaluate a power, we need to multiply the base by itself for the number of times given by the exponent. Therefore, we need to multiply 5 by itself 2 times, which is equivalent to \(5 * 5\).

Step 3: Calculate the Result

Calculating 5 times 5 gets us the result 25. Therefore, \(5^{2} = 25\).

Key Concepts

Understanding Base and Exponent

Exponentiation involves two crucial components: the base and the exponent. The base is the number you start with, in this case, 5. The exponent is the small number positioned to the upper right of the base, here it is 2. This exponent signifies the number of times the base is to be used as a factor in multiplication. In our example, since the exponent is 2, it means the base 5 will be multiplied by itself once, resulting in the expression 5 multiplied by 5. This is the fundamental principle behind exponentiation, as it allows powers to be expressed compactly.

Employing the Power Rule

Exponentiation is efficiently executed through the power rule. This rule simplifies expressions where a number is repeatedly multiplied by itself. To apply the power rule, identify the base and multiply it by itself, as many times as determined by the exponent. For the expression \(5^{2}\), the power rule guides us to multiply 5 by itself 2 times. This results in the expression \(5 \times 5\). Recognizing and applying the power rule effectively demystifies what it means to raise a number to a given power. It's a straightforward pathway to simplifying seemingly complex calculations.

Executing Multiplication

Once the power rule has been applied, it's time to perform the multiplication. For \(5^{2}\), this involves the simple calculation of 5 multiplied by 5. This is a straightforward step where basic multiplication rules apply. When you multiply 5 by 5, the outcome is 25.
Here's a quick multiplication tip involving small numbers:

• Whenever you multiply two like small numbers, the result can often be easily remembered.
• For 5's, think of half tens since 5 x 2 = 10; doubling from there isn't far.

This step solidifies the understanding of exponentiation by reinforcing the meaning of multiplying a number by itself. Remember, practicing multiplication helps build speed and confidence in handling powers of various bases.