Example Question - lcd calculation
Here are examples of questions we've helped users solve.
Simplifying Fractions with Common Denominator
To simplify the given expression, we first need to identify a common denominator for the fractions. The terms have the following denominators: 27x^2, 12, and 2(3x). The least common denominator (LCD) among these must be a number that each denominator can divide into without a remainder. The prime factorization of the denominators would be: - For 27x^2: 3^3 * x^2 - For 12: 2^2 * 3 - For 2(3x): 2 * 3 * x The LCD needs to account for the highest powers of each prime and variable present in all the denominators, which is: 2^2 * 3^3 * x^2. This simplifies to 108x^2. Now that we have identified the LCD as 108x^2, we can rewrite each fraction with this common denominator: - \( \frac{5}{27x^2} \) becomes \( \frac{5 \cdot 4}{108x^2} \) because 27x^2 times 4 equals 108x^2. - \( \frac{5}{12} \) becomes \( \frac{5 \cdot 9x^2}{108x^2} \) because 12 times 9x^2 equals 108x^2. - \( \frac{2}{2(3x)} \) becomes \( \frac{2 \cdot 18x}{108x^2} \) because 2(3x) times 18x equals 108x^2. Rewriting the expression with the common denominator: \[ \frac{5 \cdot 4}{108x^2} + \frac{5 \cdot 9x^2}{108x^2} + \frac{2 \cdot 18x}{108x^2} \] This simplifies to: \[ \frac{20}{108x^2} + \frac{45x^2}{108x^2} + \frac{36x}{108x^2} \] Now, add the fractions together, noting that the x^2 terms in the second fraction will cancel out with the x^2 in the denominator: \[ \frac{20 + 45x^2 + 36x}{108x^2} \] There are no like terms to combine, so this is as simplified as it gets without factoring the numerator, which is not suggested in this context. Also, there seem to be no common factors between the numerator and the denominator that can be cancelled out. So the final simplified expression is: \[ \frac{20 + 45x^2 + 36x}{108x^2} \]
Mathematics Problems Solution Explanation
The image displays two mathematics problems. For clarity, I will provide the complete solution for both of them: #1. Find the LCD (Least Common Denominator). The expression provided is (x - 2)(x + 1). This is not a fraction, so typically we wouldn't be looking for an LCD, which is common when dealing with fractions. However, if we interpret the expression as a product of two binomials that might serve as denominators in separate fractions, then the LCD would indeed be the product itself, which is (x - 2)(x + 1). #2. From the correct answer of #1, multiply every term by the LCD. The hint given suggests that after multiplication, the pattern should look like: x × (some expression) + (-2) × (some expression) = (-2) × (some expression) Assuming that we are to distribute (x - 2)(x + 1) across each term of some expression, which is not provided, we would do so as follows for a generic term 'a': a × (x - 2)(x + 1) = a(x^2 + x - 2x - 2) = a(x^2 - x - 2) Without additional context or an actual expression to work with, this is as far as we can solve. Each term in the original expression would be multiplied by (x - 2)(x + 1), distributed, and simplified as shown above.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com