Multiply exponents: \( m^{-3 \times 4} = m^{-12} = \frac{1}{m^{12}} \)
Answer:
\( \frac{1}{m^{12}} \)
Divide coefficients: \( \frac{12}{3} = 4 \); Powers: \( y^{6 - (-2)} = y^{6+2} = y^8 \)
Answer:
\( 4y^8 \)
Distribute power: \( 3^2 (a^2)^2 (b^{-4})^2 = 9a^4b^{-8} = \frac{9a^4}{b^8} \)
Answer:
\( \frac{9a^4}{b^8} \)
Subtract exponents: \( 2^{-4 - (-6)} = 2^{-4 + 6} = 2^2 = 4 \)
Answer:
\( 2^2 = 4 \)
Teacher Resource • Page 1 Subtotal: 18 pts Page 1 of 2
• Multi-Step, Error Analysis & Reasoning Key
Page 2 Total: 12 pts
Part 3
1.5 pts each (6 pts total)
1. Inside: \( x^{3 + (-5)} = x^{-2} \)
2. Power to power: \( (x^{-2})^2 = x^{-4} \)
3. Quotient: \( \frac{x^{-4}}{x^{-8}} = x^{-4 - (-8)} = x^4 \)
Final Form:
\( x^4 \)
1. Expand numerator: \( 2^3 a^9 b^{-6} = 8a^9b^{-6} \)
2. Divide coeff: \( \frac{8}{4} = 2 \); \( a^{9 - (-1)} = a^{10} \)
3. Variable b: \( b^{-6 - 4} = b^{-10} = \frac{1}{b^{10}} \)
Final Form:
\( \frac{2a^{10}}{b^{10}} \)
1. Numerator: \( 5^{4 + (-7)} = 5^{-3} \)
2. Denominator: \( 5^{2 \times (-2)} = 5^{-4} \)
3. Quotient: \( 5^{-3 - (-4)} = 5^1 = 5 \)
Answer:
\( 5^1 = 5 \)
1. Inside: \( 3 p^{-2-3} q^{4 - (-1)} = 3 p^{-5} q^5 \)
2. Outer power: \( 3^{-2} p^{(-5)(-2)} q^{(5)(-2)} = \frac{1}{9} p^{10} q^{-10} \)
3. Positive exponents: \( \frac{p^{10}}{9q^{10}} \)
Final Form:
\( \frac{p^{10}}{9q^{10}} \)
Part 4
1.5 pts each (3 pts total)
Jordan's Work: \( 3^4 \cdot 3^2 = 9^6 \) Base Error
1. Identified Misconception:
Error: Jordan multiplied the bases (\( 3 \times 3 = 9 \)). Under the product rule, the base remains unchanged because exponents count repeated factors of the same base.
2. Correct Answer:
\( 3^{4+2} = 3^6 = 729 \)
Taylor's Work: \( \frac{x^6}{x^{-2}} = x^{6 - 2} = x^4 \) Sign Error
1. Identified Misconception:
Error: Taylor subtracted positive \( 2 \) instead of subtracting \( -2 \). The quotient rule requires \( 6 - (-2) = 6 + 2 = 8 \).
2. Correct Answer:
\( x^{6 - (-2)} = x^{6+2} = x^8 \)
Part 5
1.5 pts each (3 pts total)
17. Conceptual Proof of \( 5^0 = 1 \)
Exemplar Proof: Consider \( \frac{5^2}{5^2} \). By division of any non-zero number by itself, \( \frac{5^2}{5^2} = \frac{25}{25} = 1 \). By the quotient property, \( \frac{5^2}{5^2} = 5^{2-2} = 5^0 \). By the transitive property, \( 5^0 = 1 \).
Rubric: 1 pt for showing quotient rule subtraction \( n - n = 0 \); 0.5 pt for connecting \( a/a = 1 \).
18. True/False Claim Evaluation
Exemplar Response: False. A negative exponent indicates a reciprocal (repeated division), not a negative value. For example, \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \), which is a positive value (\( \frac{1}{9} > 0 \)).
Rubric: 0.5 pt for "False"; 1 pt for valid counterexample demonstrating positive reciprocal value.
Teacher Resource • Total Points: 30 / 30 Page 2 of 2
Respuesta:
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Respuesta:
Matemáticas 8.º Grado • Expresiones y Ecuaciones (8.EE.A.1) Página 1 de 2
• Pasos Guiados, Análisis de Error y Razonamiento
Nombre:
Parte 3
Sin exponentes negativos en la respuesta final.
Paso 1: Paréntesis • Paso 2: Potencia • Paso 3: Dividir
Respuesta:
Paso 1: Eleva el numerador al cubo • Paso 2: Divide coeficientes y variables
Respuesta:
Paso 1: Arriba suma exponentes • Paso 2: Abajo multiplica • Paso 3: Resta
Potencia / Valor:
Paso 1: Simplifica adentro • Paso 2: Aplica el exponente \( -2 \)
Respuesta:
Parte 4
Trabajo de Jordan: ¡Hay un error!
Problema: \( 3^4 \cdot 3^2 \) • Jordan escribió: \( 9^6 \)
1. Completa la oración (Sentence frame):
Jordan multiplicó las bases \( (3 \times 3 = 9) \). El error es que la base debe quedarse igual como: ________
2. Escribe la respuesta correcta:
Trabajo de Taylor: ¡Hay un error!
Problema: \( \frac{x^6}{x^{-2}} \) • Taylor escribió: \( x^{6 - 2} = x^4 \)
1. Completa la oración (Sentence frame):
Taylor restó \( +2 \) en vez de \( -2 \). Al restar un negativo, menos por menos es más: \( 6 - (-2) = 6 + \) ________ = ________
2. Escribe la respuesta correcta:
Parte 5
17. ¿Por qué \( 5^0 = 1 \)? Prueba Visual
Completa los espacios usando la regla de división:
18. ¿Verdadero o Falso? (True / False) Evaluar Afirmación
Afirmación: "Un exponente negativo siempre da un número negativo."
Verdadero (True) Falso (False)
Ejemplo numérico (Pista: \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)):
Matemáticas 8.º Grado • Expresiones y Ecuaciones (8.EE.A.1) Página 2 de 2
7
Multiply Powers
\( (a^m)^n = a^{m \cdot n} \)
A power raised to another exponent: keep the base and multiply the exponents.
\( (5^3)^4 = 5^{3 \times 4} = 5^{12} \)
Exponents touching parentheses? Multiply them!
8
Distribute Power
\( (ab)^n = a^n b^n \)
Distribute the outer exponent to every coefficient and variable factor inside.
\( (3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6 \)
Caution: Don't forget to raise the number coefficient (\( 3^3 = 27 \))!
9
Always 1
\( a^0 = 1 \quad (a \neq 0) \)
Any non-zero quantity raised to the power of zero equals 1.
\( 8^0 = 1 \quad \text{and} \quad (-12x^5y)^0 = 1 \)
Why? \( \frac{a^3}{a^3} = 1 \), but quotient rule gives \( a^{3-3} = a^0 \)!
Regardless of size: big or small, raised to 0 equals 1!
10
Reciprocal
\( a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n \)
A negative exponent means repeated division (reciprocal), NOT a negative answer!
\( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \)
Cross the fraction line to change the sign of the exponent!
11
Fraction Inversion
Flipping a fraction upside-down so the numerator and denominator switch places.
\( \left(\frac{2}{3}\right)^{-1} = \frac{3}{2} \quad \text{and} \quad \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n \)
El recíproco invierte la fracción
Flip the fraction to flip the exponent from negative to positive!
12
Key Distinction
The regular number multiplying a variable. Multiply/divide coefficients normally; follow exponent laws for powers!
\( (2x^3)(4x^5) = (2 \cdot 4) \cdot x^{3+5} = 8x^8 \)
Coefficients multiply normally: \( 2 \times 4 = 8 \). Exponents add: \( 3 + 5 = 8 \).
Grade 8 Mathematics • Integer Exponents Reference Flashcards Sheet 2 of 2
