Perfect Cube Monomials Identifying Numbers For Transformation In $125x^{18}y^3z^{25}$

In the realm of mathematics, perfect cubes hold a significant position, often appearing in various algebraic expressions and equations. Understanding the properties of perfect cubes is crucial for simplifying and solving complex mathematical problems. This article delves into the concept of perfect cubes, specifically focusing on identifying the number within a given monomial that needs adjustment to transform the entire expression into a perfect cube. We will dissect the monomial $125x^{18}y^3z^{25}$, analyze each component, and determine which exponent requires modification. By the end of this exploration, you will not only grasp the solution to this particular problem but also gain a deeper understanding of perfect cubes and their applications in mathematics.

Understanding Perfect Cubes

To effectively address the question of which number in the monomial $125x^{18}y^3z^{25}$ needs to be changed to make it a perfect cube, it's essential to first establish a clear understanding of what perfect cubes are. A perfect cube is a number that can be obtained by cubing an integer, meaning it is the result of multiplying an integer by itself three times. Mathematically, a number n is a perfect cube if there exists an integer k such that n = k³. Examples of perfect cubes include 1 (1³ = 1), 8 (2³ = 8), 27 (3³ = 27), 64 (4³ = 64), and 125 (5³ = 125). Recognizing perfect cubes is a fundamental skill in algebra and number theory, as it often simplifies calculations and aids in solving equations. In the context of monomials, a perfect cube monomial is one where the coefficient is a perfect cube and the exponents of all the variables are multiples of 3. This ensures that the entire monomial can be expressed as the cube of another monomial. Understanding this concept is crucial for the task at hand, which involves identifying which part of the given monomial needs adjustment to fit this criterion. We will apply this understanding to the monomial $125x^{18}y^3z^{25}$, examining each component to determine the necessary modification.

Perfect Cube Coefficients

When dealing with monomials and determining if they are perfect cubes, the coefficient plays a crucial role. The coefficient, which is the numerical factor in the monomial, must itself be a perfect cube for the entire expression to be a perfect cube. This means the coefficient should be the result of cubing an integer. For instance, in the monomial $125x^{18}y^3z^{25}$, the coefficient is 125. To ascertain if 125 is a perfect cube, we need to determine if there is an integer that, when cubed, equals 125. In this case, $5^3 = 5 * 5 * 5 = 125$, which confirms that 125 is indeed a perfect cube. Therefore, the coefficient in this monomial already satisfies the condition for being a perfect cube. However, in other scenarios, if the coefficient were not a perfect cube, it would need to be adjusted to the nearest perfect cube value to transform the entire monomial into a perfect cube. Understanding how to identify and manipulate coefficients to achieve perfect cube status is a fundamental skill in algebra, particularly when simplifying expressions and solving equations involving radicals and exponents. This initial check of the coefficient allows us to narrow our focus to the variable exponents in the monomial.

Perfect Cube Exponents

Beyond the coefficient, the exponents of the variables are equally critical in determining whether a monomial is a perfect cube. For a monomial to be a perfect cube, the exponent of each variable must be a multiple of 3. This is because when taking the cube root of a variable term (e.g., $x^n$), the exponent is divided by 3. If the exponent is a multiple of 3, the result will be an integer, indicating a perfect cube. Consider the monomial $125x^{18}y^3z^{25}$. The exponents of the variables are 18, 3, and 25. To determine if these exponents meet the criteria for a perfect cube, we need to check if they are divisible by 3. The exponent 18 is divisible by 3 ($18 ÷ 3 = 6$), and the exponent 3 is also divisible by 3 ($3 ÷ 3 = 1$). However, the exponent 25 is not divisible by 3. This means that the term $z^{25}$ is not a perfect cube. To make the monomial a perfect cube, we need to change the exponent of z to the nearest multiple of 3. This understanding of exponent divisibility is a cornerstone of simplifying expressions and working with radicals, as it allows us to identify and manipulate terms to achieve perfect cube status. Therefore, our attention now turns to modifying the exponent 25 to the nearest multiple of 3.

Analyzing the Monomial $125x^{18}y^3z^{25}$

To pinpoint the exact number that needs modification in the monomial $125x^{18}y^3z^{25}$ to render it a perfect cube, we must meticulously examine each component of the expression. As established earlier, a perfect cube monomial requires both the coefficient to be a perfect cube and the exponents of all variables to be multiples of 3. In this monomial, the coefficient is 125, which we have already confirmed is a perfect cube ($5^3 = 125$). Now, let's turn our attention to the exponents of the variables. The variables are x, y, and z, with exponents 18, 3, and 25, respectively. We need to determine which of these exponents, if any, are not multiples of 3. As previously discussed, 18 is divisible by 3, as $18 ÷ 3 = 6$. Similarly, 3 is divisible by 3, as $3 ÷ 3 = 1$. However, 25 is not divisible by 3. When 25 is divided by 3, the result is 8 with a remainder of 1. This indicates that $z^{25}$ is the component preventing the entire monomial from being a perfect cube. To rectify this, we need to adjust the exponent 25 to the nearest multiple of 3, which will transform the monomial into a perfect cube. This detailed analysis of each component of the monomial allows us to focus our efforts on the specific element that requires modification.

Identifying the Number to Change

Having established that the monomial $125x^{18}y^3z^{25}$ is not a perfect cube due to the exponent of the variable z, the next step is to precisely identify which number needs to be changed. As we've determined, the coefficient 125 is a perfect cube, and the exponents 18 and 3 of variables x and y respectively, are multiples of 3. This leaves us with the exponent 25 of the variable z. The exponent 25 is the number that needs adjustment to make the monomial a perfect cube. To achieve this, we need to find the nearest multiple of 3 to 25. The multiples of 3 around 25 are 24 and 27. We can choose either 24 or 27 as the new exponent, depending on whether we want to decrease or increase the value of the term. If we choose 24, the term becomes $z^{24}$, and if we choose 27, the term becomes $z^{27}$. Both $z^{24}$ and $z^{27}$ are perfect cubes because their exponents are divisible by 3. However, without further context or specific instructions, the most straightforward approach is to change 25 to either 24 or 27. Therefore, the number that needs to be changed in the monomial to make it a perfect cube is 25, the exponent of the variable z. This focused identification is crucial for effectively transforming the monomial into its perfect cube form.

Determining the Correct Option

Now that we have identified that the number 25, the exponent of z in the monomial $125x^{18}y^3z^{25}$, needs to be changed to make the expression a perfect cube, we can directly address the multiple-choice options provided. The options are:

• A. 3
• B. 18
• C. 25
• D. 125

Based on our analysis, the correct answer is C. 25, as it is the exponent of z that prevents the monomial from being a perfect cube. The other options can be eliminated as follows:

• A. 3 is the exponent of y, and since 3 is a multiple of 3, the term $y^3$ is already a perfect cube.
• B. 18 is the exponent of x, and since 18 is a multiple of 3, the term $x^{18}$ is also a perfect cube.
• D. 125 is the coefficient of the monomial, and as we established earlier, 125 is a perfect cube ($5^3 = 125$).

Therefore, by process of elimination and our detailed analysis, we can confidently conclude that option C, 25, is the correct answer. This straightforward approach, combining theoretical understanding with practical application, is key to solving similar mathematical problems efficiently and accurately.

Conclusion

In summary, the question of which number in the monomial $125x^{18}y^3z^{25}$ needs to be changed to make it a perfect cube has been thoroughly addressed. Through a step-by-step analysis, we established the criteria for perfect cube monomials, examined each component of the given expression, and identified that the exponent 25 of the variable z is the number requiring modification. This conclusion aligns with the correct option, C. 25, among the provided choices. Understanding perfect cubes and their properties is not only essential for solving this specific problem but also for tackling a wide range of algebraic and mathematical challenges. The ability to identify and manipulate perfect cubes simplifies complex expressions, aids in solving equations, and enhances overall mathematical proficiency. This comprehensive guide has hopefully provided a clear and detailed explanation, empowering you to confidently approach similar problems in the future. By mastering these fundamental concepts, you can unlock a deeper appreciation for the elegance and logic inherent in mathematics.