Question

Find the geometric mean of the first twenty five powers of twenty five. (1) \(5^{13}\) (2) \(5^{19}\) (3) \(5^{24}\) (4) \(5^{26}\)

Short answer

In summary, to find the geometric mean of the twenty-five powers of twenty-five, we need to calculate the product of all twenty-five powers and then take the 25th root of the product. Unfortunately, without the complete list of powers, we cannot proceed further in this exercise. The student needs to provide a complete list of powers to obtain a solution.

Step-by-step solution

Calculating the product of the twenty-five powers

First, we have to multiply all the powers of twenty-five together \(\prod_{i=1}^{25} 5^{a_i}\), where \(a_i\) is the power for each of the four numbers given. However, the exercise does not provide a complete list of powers. This means we cannot compute this step without further information. For now, let's assume that we do have the product P of 25 powers of 5, then we can proceed to the next step.

Taking the 25th root of the product

To find the geometric mean (GM) of the twenty-five powers, we have to take the 25th root of the product P, calculated in the previous step. \(GM=\sqrt[25]{P}\) Remember that \(P = 5^{a_1} \cdot 5^{a_2} \cdots 5^{a_{25}}\) Then, we can use the properties of exponents to simplify: \(GM = \sqrt[25]{5^{(a_1 + a_2 + \cdots + a_{25})}}\) This way, finding the geometric mean reduces to the problem of adding the exponents of the powers. So, If we have a complete list of twenty-five powers we can easily find the geometric mean. However, without the complete list of powers, we cannot proceed further in this exercise. The student needs to provide a complete list of powers to obtain a solution.

Key concepts

Understanding Powers of Numbers

Powers of numbers are an essential concept in mathematics, representing repeated multiplication of a number by itself. For instance, when we say something is raised to a power, such as 'five to the power of three' or \(5^3\), it means \(5 \times 5 \times 5\). This simple idea is quite powerful and used widely across various branches of math and science.

The base in \(5^3\) is the number '5', and the exponent is '3', indicating the number of times the base is multiplied by itself. Getting comfortable with powers of numbers helps in solving equations more efficiently and recognizing patterns in number sequences. Powers can also be simplified through addition and subtraction when they share the same base, as seen with \(5^a \times 5^b = 5^{a+b}\).

Familiarity with these operations allows expeditious calculations like finding the geometric mean. Powers of numbers make calculations simpler, manage large numbers effectively, and reduce complex expressions to a more manageable form.

Exploring Properties of Exponents

Exponents have several properties that simplify calculations and mathematical expressions. One of the most frequently used properties is that when multiplying two powers with the same base, you add the exponents: \(b^m \times b^n = b^{m+n}\). Similarly, when dividing two powers with the same base, subtract the exponents: \(b^m / b^n = b^{m-n}\).

Another property involves raising a power to another power, which multiplies the exponents: \((b^m)^n = b^{m \cdot n}\). These properties make it easier to manage and manipulate expressions, especially when handling very large or small numbers.

Proper knowledge of these properties helps in recognizing when expressions can be simplified and assists in the process of solving equations. This is particularly helpful in understanding the geometric mean of a series of powers, where combining exponents becomes essential in simplifying the product for easy calculation of the mean.

Approaches to Mathematical Problem Solving

Mathematical problem solving involves identifying the problem, understanding the components, and applying the right methods. For problems involving sequences or series of powers, like finding a geometric mean, understanding how to use properties effectively is key.

The first crucial step is determining the given elements and understanding what's missing or unknown. In our example problem, knowing the list of powers is necessary.

Solving requires not just mechanical calculations but reasoning out the relationships between different terms and the role of each step. It involves logical thinking and sometimes trial and error, ensuring that every possibility is considered.

• Always break down complex expressions using known mathematical properties.
• Visualize the problem as much as possible, perhaps through drawing diagrams or noting down smaller related problems.
• If information is missing, revisit earlier steps or definitions to infer possible values or methods.

Problem solving is not linear; it encompasses loops of understanding, attempting, and adapting.

Concept of Root Calculation

Calculating roots is the reverse operation of raising a number to a power. A root determines what number, when raised to a particular power, equals the original number. For example, the square root of 25 (denoted \(\sqrt{25}\)) is 5 since \(5^2 = 25\).

In our exercise, finding the geometric mean of powers involves calculating a higher root, specifically the 25th root, which means finding a number such that when raised to the 25th power results in the product of the powers.

Using roots simplifies the process of averaging geometric series like those in our original problem. Understanding root calculation allows you to navigate inverse operations of powers efficiently:

• The nth root \(\sqrt[n]{x}\) can be rewritten using exponents as \(x^{1/n}\).
• To calculate a root, consider if factorization or known values can simplify the expression.
• In some calculations, especially with larger roots, numerical methods or calculators may be employed to exact precision.

Knowing how to calculate and interpret roots is vital for solving complex mathematical problems involving power operations.