The product of two non-zero expressions is (x + y + z) p 3. If their HCF is p 2, then their LCM is
(x + y + z) p
Understanding the Relationship Between Product, HCF, and LCM
In algebra, there's a fundamental relationship between two non-zero expressions, their Highest Common Factor (HCF), and their Lowest Common Multiple (LCM). This relationship is key to solving problems like finding the LCM when the product and HCF are known.
The Fundamental Relationship: Product = HCF \(\times\) LCM
For any two non-zero expressions, say A and B, their product is equal to the product of their HCF and LCM. This can be written as:
\( \text{Product of A and B} = \text{HCF(A, B)} \times \text{LCM(A, B)} \)
This property holds true for both numbers and algebraic expressions.
Applying the Relationship to Find the LCM
The question provides us with the product of two non-zero expressions and their HCF. We are asked to find their LCM.
- Product of the two expressions = \((x + y + z) p^3\)
- HCF of the two expressions = \(p^2\)
- Let the LCM of the two expressions be L.
Using the relationship formula:
\( \text{Product} = \text{HCF} \times \text{LCM} \)
Substitute the given values into the formula:
\( (x + y + z) p^3 = p^2 \times L \)
Solving for the LCM (L)
To find L, we need to isolate it. We can do this by dividing both sides of the equation by the HCF, which is \(p^2\):
\( L = \frac{(x + y + z) p^3}{p^2} \)
Now, we simplify the expression on the right side. We can separate the terms:
\( L = (x + y + z) \times \frac{p^3}{p^2} \)
Recall the rule of exponents for division: \( \frac{a^m}{a^n} = a^{m-n} \)\(. Applying this rule to \)\( \frac{p^3}{p^2} \), we get:
\( \frac{p^3}{p^2} = p^{3-2} = p^1 = p \)
Substitute this simplified term back into the equation for L:
\( L = (x + y + z) \times p \)
So, the LCM of the two expressions is \((x + y + z) p\).
Conclusion: The LCM Value
Based on the calculation, the Lowest Common Multiple of the two non-zero expressions is \((x + y + z) p\).
Revision Table: HCF and LCM Relationship
| Concept | Description | Relationship |
|---|---|---|
| Product of Two Expressions (A and B) | The result of multiplying Expression A by Expression B. | Product = HCF \(\times\) LCM |
| HCF (Highest Common Factor) | The largest expression that divides both A and B without leaving a remainder. | |
| LCM (Lowest Common Multiple) | The smallest expression that is a multiple of both A and B. | LCM = Product / HCF |
Additional Information: HCF and LCM in Algebra
Finding the HCF and LCM of algebraic expressions involves factoring the expressions into their prime factors (like we factor numbers). The HCF is the product of the common factors raised to the lowest power. The LCM is the product of all unique factors raised to the highest power.
For example, consider the expressions \(6a^2b\)\( and \)\(9ab^3\).
- Factor \(6a^2b\)\(: \)\(2 \times 3 \times a^2 \times b\)
- Factor \(9ab^3\)\(: \)\(3^2 \times a \times b^3\)
To find the HCF:
- Common factors are 3, a, and b.
- Lowest power of 3 is \(3^1\).
- Lowest power of a is \(a^1\).
- Lowest power of b is \(b^1\).
- HCF = \(3^1 \times a^1 \times b^1 = 3ab\)
To find the LCM:
- All unique factors are 2, 3, a, and b.
- Highest power of 2 is \(2^1\).
- Highest power of 3 is \(3^2\).
- Highest power of a is \(a^2\).
- Highest power of b is \(b^3\).
- LCM = \(2^1 \times 3^2 \times a^2 \times b^3 = 2 \times 9 \times a^2 \times b^3 = 18a^2b^3\)
Now, let's check the product relationship:
- Product of expressions = \((6a^2b)(9ab^3) = 54a^3b^4\)
- HCF \(\times\) LCM = \((3ab)(18a^2b^3) = 54a^3b^4\)
The relationship holds true: Product = HCF \(\times\) LCM.
This example reinforces the concept used to solve the problem, demonstrating how the product, HCF, and LCM are related in algebraic expressions.
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