Example Question - distributive property
Here are examples of questions we've helped users solve.
Distributing Variables in Algebraic Expressions
<p>1. Distribute \(3\) in \(3(4-x)\):</p> <p>\(3 \cdot 4 - 3 \cdot x = 12 - 3x\)</p> <p>2. Distribute \(4\) in \(4(y-2)\):</p> <p>\(4 \cdot y - 4 \cdot 2 = 4y - 8\)</p> <p>3. Distribute \(5\) in \(5/(2(3+7))\):</p> <p>Since \(5\) is already outside the parenthesis, we focus on the denominator:</p> <p>Calculate \(2(3+7) = 2 \cdot 10 = 20\), hence the expression becomes:</p> <p>\(\frac{5}{20} = \frac{1}{4}\)</p>
Distributive Property in Arithmetic
<p>Note that the equation uses the distributive property of multiplication over addition.</p> <p>Given: \(6 \times 8 = (5 + 1) \times 8\)</p> <p>Distribute \(8\) across \(5 + 1\):</p> <p>\(6 \times 8 = (5 \times 8) + (1 \times 8)\)</p> <p>\(6 \times 8 = 40 + 8\)</p> <p>\(6 \times 8 = 48\)</p> <p>Therefore, the solution completes the equation: \(6 \times 8 = 48\).</p>
Exploring the Distributive Property of Multiplication
<p>The image shows a multiplication problem being solved using the distributive property of multiplication over addition:</p> <p>\[ 6 \times 9 = 6 \times (5 + 4) \]</p> <p>\[ = (6 \times 5) + (6 \times 4) \]</p> <p>\[ = 30 + 24 \]</p> <p>\[ = 54 \]</p>
Simplifying Expressions Using the Distributive Property
1. Apply the distributive property to 3p(p - q): 3p * p - 3p * q = 3p^2 - 3pq 2. Expand the square of the binomial (2p - q)^2: (2p - q)(2p - q) = 4p^2 - 4pq + q^2 3. Subtract the expanded binomial from the first expression: (3p^2 - 3pq) - (4p^2 - 4pq + q^2) 4. Distribute the negative sign to each term in the second expression: 3p^2 - 3pq - 4p^2 + 4pq - q^2 5. Combine like terms: -p^2 + pq - q^2 So the simplified expression is: -p^2 + pq - q^2
Complex Number Multiplication
The image shows a handwritten problem which asks to compute the product of the following complex numbers: 1) \( Z_1 = 1 + 3i \) and \( Z_2 = 5 + i \) 2) \( Z_1 = 3 - 4i \) and \( Z_2 = 3 + 2i \) To compute the product of two complex numbers, you multiply them using the distributive property (also known as the FOIL method for binomials), which stands for "First, Outer, Inner, Last." Also remember that \( i^2 = -1 \). Let's solve the first product: \( Z_1 \cdot Z_2 = (1 + 3i)(5 + i) \) 1. First (1 * 5): 5 2. Outer (1 * i): i 3. Inner (3i * 5): 15i 4. Last (3i * i): 3i^2 Combining these we get: \( 5 + i + 15i + 3i^2 \) Since \( i^2 = -1 \), substitute for \( i^2 \): \( 5 + i + 15i + 3(-1) \) \( 5 + 16i - 3 \) Combine like terms: \( (5 - 3) + 16i \) \( 2 + 16i \) So the product of the first pair is \( 2 + 16i \). Now for the second product: \( Z_1 \cdot Z_2 = (3 - 4i)(3 + 2i) \) 1. First (3 * 3): 9 2. Outer (3 * 2i): 6i 3. Inner (-4i * 3): -12i 4. Last (-4i * 2i): -8i^2 Now combine these: \( 9 + 6i - 12i - 8i^2 \) Substitute \( i^2 \) with -1: \( 9 + 6i - 12i - 8(-1) \) \( 9 - 6i - 8 \) Combine like terms: \( (9 + 8) - 6i \) \( 17 - 6i \) So the product of the second pair is \( 17 - 6i \).
Solving Expression using Distributive Property and Combining Like Terms
To solve the expression given in the image, you'll need to apply the distributive property and combine the like terms. The expression is: \(-6x^2(3x^5)\) When multiplying two exponential expressions with the same base, you add the exponents: \(x^2 \cdot x^5 = x^{2+5} = x^7\) Now multiply the coefficients, which are the numerical parts: \(-6 \cdot 3 = -18\) Combining both parts, you get: \(-18x^7\) So the simplified expression is: \(-18x^7\)
Expanding Algebraic Expression Using FOIL Method
The image shows an algebraic expression that needs to be expanded. To solve the expression (2x + 1)(3x + 2), we use the distributive property (also known as the FOIL method for binomials), where we multiply each term in the first parenthesis by each term in the second parenthesis. Here’s how it is expanded: (2x + 1)(3x + 2) = 2x * 3x + 2x * 2 + 1 * 3x + 1 * 2 Now, multiply the terms: = 6x^2 + 4x + 3x + 2 Combine like terms: = 6x^2 + 7x + 2 So, the expanded form of the expression is: 6x^2 + 7x + 2
Solving Surds Expression with Distributive Property
The given expression is: \[ \sqrt{a - 2} \times (\sqrt{a} + 2) \] To solve it, we can apply the distributive property, which states that \[ a(b + c) = ab + ac \]. Here, we'll multiply each term inside the parenthesis by \(\sqrt{a - 2}\): \[ \sqrt{a - 2} \times \sqrt{a} + \sqrt{a - 2} \times 2 \] Now, multiply the terms individually: 1. \( \sqrt{a - 2} \times \sqrt{a} \) gives us \( \sqrt{(a - 2) \times a} \), which simplifies to \( \sqrt{a^2 - 2a} \). However, since we are multiplying roots with the same index, we can directly combine them under a single root: \( \sqrt{a^2 - 2a} = a\sqrt{1} - 2\sqrt{a} \), which simplifies to \( a - 2\sqrt{a} \). This simplification took an incorrect step, as it is not valid to separate the terms inside the square root when they are under multiplication. The correct simplification is actually the square root of the product, so \(\sqrt{a(a - 2)}\) is \( \sqrt{a^2 - 2a} \) and should remain under the square root sign. 2. \( \sqrt{a - 2} \times 2 \) gives us \( 2\sqrt{a - 2} \). Combine these results and we get the correct expression: \[ a - 2\sqrt{a} + 2\sqrt{a - 2} \] However, to simplify further, we need to assume that this is a multiplication of conjugate surds which would yield a simplified expression without any square roots: \[ (\sqrt{a} - 2)(\sqrt{a} + 2) \] Now let's multiply these terms: \[ = a + 2\sqrt{a} - 2\sqrt{a} - 4 \] The terms \( 2\sqrt{a} \) and \( -2\sqrt{a} \) cancel each other out: \[ = a - 4 \] The correct answer to the expression is \( a - 4 \), which corresponds to option C.
Solving Expressions Using Distributive Property
To solve the expression given in the image, you'll need to apply the distributive property, which states that \( a(b + c) = ab + ac \). So for the expression \( 4(8n + 2) \), use the distributive property as follows: \( 4 \times 8n + 4 \times 2 \) Now multiply the numbers: \( 32n + 8 \) The simplified form of the expression is \( 32n + 8 \).
Solving an Algebraic Expression Using the Distributive Property
To solve the expression \( 6(1 - 5m) \), you need to apply the distributive property. This means that you'll multiply the number 6 by each term inside the parentheses. So, firstly, multiply 6 by 1, which gives you 6. Then, multiply 6 by \(-5m\), which gives you \(-30m\). Combine these two results to get the simplified expression: \[ 6 - 30m \]
Solving Expression Using Distributive Property
To solve the expression given in the image, you need to apply the distributive property (also known as the FOIL method for binomials), which states that a(b + c) = ab + ac. Given the expression: (-7x - 8)(12 + 8x), we apply the distributive property as follows: (-7x * 12) + (-7x * 8x) + (-8 * 12) + (-8 * 8x) Now perform the multiplications: (-84x) + (-56x^2) - (96) - (64x) Combine like terms: -56x^2 - 84x - 64x - 96 -56x^2 - 148x - 96 The final simplified expression is: -56x^2 - 148x - 96
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