Example Question - binomial multiplication
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Algebraic Expression Simplification
<p>Para resolver la expresión algebraica dada, se debe aplicar la propiedad distributiva (ley distributiva).</p> <p>La propiedad distributiva establece que \(a(b+c) = ab + ac\). En este caso, tendremos términos similares con \(x^3 \times x^3\) y \(x^3 \times (-x^7)\).</p> <p>Para simplificar, seguimos estos pasos:</p> <p>\((x^3)(x^3 - x^7) = x^3 \times x^3 - x^3 \times x^7\)</p> <p>Utilizamos la ley de los exponentes que dice que \(x^a \times x^b = x^{a+b}\):</p> <p>\(x^3 \times x^3 = x^{3+3} = x^6\)</p> <p>\(x^3 \times (-x^7) = -x^{3+7} = -x^{10}\)</p> <p>Combinando estos términos, la expresión se simplifica a:</p> <p>\(x^6 - x^{10}\)</p>
Algebraic Expression Simplification
The image shows an algebraic expression that requires simplification: (2√x - √y)(6√x + 5√y) This is a binomial multiplication problem, which we can solve using the FOIL method, where FOIL stands for "First, Outer, Inner, Last." This means we multiply the first terms in each binomial, then the outer terms, followed by the inner terms, and finally the last terms, combining like terms where possible. Here's how we apply FOIL to this problem: First: (2√x)*(6√x) = 12x Outer: (2√x)*(5√y) = 10√xy Inner: (-√y)*(6√x) = -6√xy Last: (-√y)*(5√y) = -5y Combining these results, we get: 12x + 10√xy - 6√xy - 5y Combine like terms: 12x + (10√xy - 6√xy) - 5y 12x + 4√xy - 5y The simplified form of the expression is: 12x + 4√xy - 5y
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.
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