Example Question - expressions
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Understanding Exponential Expressions
<p>To simplify the expression, we use the property of exponents that states:</p> <p>When dividing like bases, you subtract the exponents:</p> <p>\(\frac{x^a}{x^b} = x^{a-b}\)</p> <p>Thus, the equation holds:</p> <p>\(\frac{x^a}{x^b} = x^{a-b}\)</p>
Mathematical Statements Evaluation
<p>Para la opción a:</p> <p>6 + 4 = 10 y 9 - 4 = 5;</p> <p>Ambas afirmaciones son verdaderas.</p> <p>Para la opción b:</p> <p>8/2 = 4 y 8 + 2 = 12;</p> <p>Ambas afirmaciones son verdaderas.</p> <p>Para la opción c:</p> <p>No se puede evaluar como verdadera o falsa ya que es una afirmación.</p> <p>Para la opción d:</p> <p>Si 3 * 7 = 21, entonces 9 - 7 = 2;</p> <p>Ambas afirmaciones son verdaderas.</p>
Simplifying Mathematical Expressions
<p>a) \left| \frac{2}{3} \cdot \left( -\frac{3}{5} \right) \cdot \frac{1}{4} \right| = \left| -\frac{2 \cdot 3}{3 \cdot 5 \cdot 4} \right| = \frac{6}{60} = \frac{1}{10}</p> <p>b) \left| \frac{1}{2} \cdot \left( \frac{2}{7} \right)^{\frac{3}{2}} \right| = \frac{1}{2} \cdot \frac{2^{\frac{3}{2}}}{7^{\frac{3}{2}}} = \frac{\sqrt{8}}{2\sqrt{343}} = \frac{2\sqrt{2}}{14\sqrt{7}} = \frac{\sqrt{2}}{7\sqrt{7}}</p> <p>c) \left| \left( \frac{5}{4} \right)^{-1} \right| = \left| \frac{4}{5} \right| = \frac{4}{5}</p>
Simplifying Algebraic Expressions
<p>Para simplificar la expresión dada:</p> <p>Comenzamos con:</p> <p> \(\frac{x^2 \cdot (y^5)^3 \cdot \left( \frac{1}{z} \right)^2}{\frac{x^3}{z^5} \cdot \left( \frac{y^2}{z} \right)^{-4}}\)</p> <p>Reescribiendo y simplificando paso a paso:</p> <p>Numerador:</p> <p> \(x^2 \cdot y^{15} \cdot \frac{1}{z^2}\)</p> <p>Denominador:</p> <p> \(\frac{x^3}{z^5} \cdot \frac{z^4}{y^8}\)</p> <p>Combinando:</p> <p> \(\frac{x^2 \cdot y^{15} \cdot z^4}{x^3 \cdot y^8 \cdot z^2 \cdot z^5}\)</p> <p>Reduciendo términos:</p> <p> \(= \frac{y^{15 - 8}}{z^{2 + 5 - 4} \cdot x^{3 - 2}}\)</p> <p>Resultado final:</p> <p> \(\frac{y^7}{z^5 \cdot x}\)</p>
Calculation of Expressions
<p>Para resolver la expresión dada:</p> <p> \( E = 100^{3^2} \cdot 25 \cdot 8^{3 - 1} + \left( \frac{1}{81} \right)^{-16} \cdot 0.29^3 \) </p> <p>Primero, cálculos intermedios:</p> <p>100 es \( 10^2 \), entonces \( 100^{3^2} = (10^2)^{9} = 10^{18} \)</p> <p>25 es \( 5^2 \), entonces \( 25 \) permanece igual.</p> <p>8 es \( 2^3 \), entonces \( 8^{3-1} = 8^2 = 64 \) o \( (2^3)^2 = 2^6 = 64 \).</p> <p>Ahora, \( E = 10^{18} \cdot 25 \cdot 64 \)</p> <p>Ahora, calcularemos \( ( \frac{1}{81} )^{-16} = 81^{16} \) y \( 0.29^3 \).</p> <p>Finalmente, sumamos ambos resultados para encontrar \( E \).</p>
Simplifying Algebraic Expressions
<p>For the first expression, \(7h + 21\), you can factor out the common factor:</p> <p>Factorization yields \(7(h + 3)\).</p> <p>For the second expression, \(6x - 12\), you can also factor out the common factor:</p> <p>Factorization yields \(6(x - 2)\).</p>
Algebraic Expression Evaluation
<p>For the expressions given:</p> <p>1. \(4(y-2) = 4y - 8\)</p> <p>2. \(5f(2f+7) = 10f^2 + 35f\)</p> <p>3. \((5x-4)(2x+3) = 10x^2 + 15x - 8x - 12 = 10x^2 + 7x - 12\)</p>
Algebraic Expressions Evaluation
<p>To evaluate the expressions:</p> <p>1. \( 4(y-2) = 4y - 8 \)</p> <p>2. \( 5f(2f+7) = 10f^2 + 35f \)</p> <p>3. \( (5x-4)(2x+3) = 10x^2 + 15x - 8 \)</p>
Simplifying a Mathematical Expression
<p>First, express each term correctly:</p> <p>1. Convert mixed number \(1 \frac{296}{36}\) to improper fraction: \( \frac{296 + 36}{36} = \frac{332}{36} \)</p> <p>2. Perform the operation \( \frac{332}{36} \div 36 \): this becomes \( \frac{332}{36 \times 36} = \frac{332}{1296} \)</p> <p>3. Multiply by \( \frac{1}{216} \): \( \frac{332}{1296} \times \frac{1}{216} = \frac{332}{1296 \times 216} \)</p> <p>4. Finally, simplify \( \frac{332}{1296 \times 216} \) to lowest terms if possible.</p>
Simplifying Algebraic Expressions
<p>First, expand the numerator:</p> <p>(m^6 n)^2 = m^{12} n^2</p> <p>Now the expression becomes:</p> <p>\frac{m^{12} n^2}{m^3 n^5}</p> <p>Next, apply the quotient rule:</p> <p>\frac{m^{12}}{m^3} = m^{12-3} = m^9</p> <p>\frac{n^2}{n^5} = n^{2-5} = n^{-3}</p> <p>The expression simplifies to:</p> <p>m^9 n^{-3}</p> <p>In positive exponent form, it is:</p> <p>\frac{m^9}{n^3}</p>
Simplifying Exponential Expressions
<p>To simplify \(10^{-8}\), we can express it as:</p> <p>\(10^{-8} = \frac{1}{10^{8}}\)</p>
Determining Polynomial Expressions
<p>(a) \( x^2 - 5 \) is a quadratic expression because it can be written in the form \( ax^2 + bx + c \) with \( a = 1, b = 0, c = -5 \).</p> <p>(b) \( 2x^2 + x \) is a quadratic expression because it can be expressed as \( ax^2 + bx + c \) with \( a = 2, b = 1, c = 0 \).</p> <p>(c) \( 3x^2 - 3x + 1 \) is a quadratic expression since it conforms to the standard form \( ax^2 + bx + c \) with \( a = 3, b = -3, c = 1 \).</p> <p>(d) \( x^4 - 2x - 1 \) is not a quadratic expression as it has a degree of 4, which exceeds 2.</p>
Identifying Variable Types in Algebraic Expressions
<p>For the expression \( x^2 - 5 \):</p> <p>This is a polynomial expression where \( x \) is a variable.</p> <p>For the expression \( \frac{1}{m^2} \):</p> <p>This is a rational expression, where \( m \) is also a variable.</p>
Basic Arithmetic Operations
<p>Para resolver la primera expresión:</p> <p>7 + 54 ÷ 3</p> <p>Primero, resuelve la división:</p> <p>54 ÷ 3 = 18</p> <p>Luego, suma el resultado a 7:</p> <p>7 + 18 = 25</p> <p>Así que, la solución es \(25\).</p> <p>Para resolver la segunda expresión:</p> <p>3 + 9 × 5</p> <p>Primero, resuelve la multiplicación:</p> <p>9 × 5 = 45</p> <p>Luego, suma el resultado a 3:</p> <p>3 + 45 = 48</p> <p>Así que, la solución es \(48\).</p>
Factoring Polynomial Expressions
Chúng ta hãy giải quyết bài toán số \( 2 \): <p>\( (x+1)(x+2)(x+3)(x+4) - 24 \)</p> <p>Bước 1: Nhận ra \( (x+1)(x+4) \) và \( (x+2)(x+3) \) là hai cặp số hạng liền kề của một dãy số liên tiếp.</p> <p>\( (x+1)(x+4) = x^2 + 5x + 4 \)</p> <p>\( (x+2)(x+3) = x^2 + 5x + 6 \)</p> <p>Bước 2: Nhân hai biểu thức trên.</p> <p>\( (x^2 + 5x + 4)(x^2 + 5x + 6) \)</p> <p>Vì cách làm trên khá dài và phức tạp, ta thử nhận ra một mẫu số chuẩn hóa:</p> <p>Bước 3: Phát hiện \( (x^2 + 5x + 4)(x^2 + 5x + 6) \) gần giống \( (x^2 + 5x + 5)^2 \), nhưng cần trừ đi \( 1 \).</p> <p>\( (x^2 + 5x + 4)(x^2 + 5x + 6) = (x^2 + 5x + 5)^2 - 1^2 \)</p> <p>Bước 4: Sử dụng công thức hiệu của hai bình phương.</p> <p>\( (x^2 + 5x + 5 + 1)(x^2 + 5x + 5 - 1) \)</p> <p>\( (x^2 + 5x + 6)(x^2 + 5x + 4) \)</p> <p>Bước 5: So sánh với biểu thức gốc và điều chỉnh để có 24 ở cuối.</p> <p>\( ((x^2 + 5x + 6)(x^2 + 5x + 4)) - 24 \)</p> <p>Bước 6: Đặt \( A = x^2 + 5x + 6 \) và \( B = x^2 + 5x + 4 \) để dễ quản lý.</p> <p>\( (AB) - 24 = (A - 2)(B + 2) \)</p> <p>Đây là kết quả sau khi đã nhân và thực hiện phép trừ.</p>
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