Question

10-121. work with your team to write four exponent problems, each having a simplification of ( x^{12} ). at least one problem must involve multiplication, one must involve grouping, and one must involve division. be creative!
10-122. gerardo is simplifying expressions with very large exponents. he arrives at each of the results below. for each result, decide if he is correct and justify your answer using the meaning of exponents.
( \boldsymbol{\text{a.}}quad \frac{x^{150}}{x^{50}} rightarrow x^3 )
( \boldsymbol{\text{b.}}quad y^{20} cdot y^{41} rightarrow y^{61} )
( \boldsymbol{\text{c.}}quad (2m^2n^{15})^3 rightarrow 2m^6n^{45} )

Explanation:

For 10-121:

1. Multiplication problem: Use the product rule of exponents ($x^a \cdot x^b = x^{a+b}$), choose exponents that add to 12.
2. Division problem: Use the quotient rule of exponents ($\frac{x^a}{x^b} = x^{a-b}$), choose exponents that subtract to 12.
3. Grouping problem: Use the power of a power rule ($(x^a)^b = x^{a \cdot b}$), choose exponents that multiply to 12.
4. Combined operations problem: Mix rules for creativity, ensuring the final result is $x^{12}$.

Step1: Evaluate part a (quotient rule)

Quotient rule: $\frac{x^a}{x^b}=x^{a-b}$. Calculate $150-50=100$, so $\frac{x^{150}}{x^{50}}=x^{100}
eq x^3$.

Step2: Evaluate part b (product rule)

Product rule: $y^a \cdot y^b=y^{a+b}$. Calculate $20+41=61$, so $y^{20} \cdot y^{41}=y^{61}$.

Step3: Evaluate part c (power of product rule)

Power of product rule: $(ab)^n=a^n b^n$. Calculate $2^3=8$, so $(2m^2n^{15})^3=8m^6n^{45}
eq 2m^6n^{45}$.

Answer:

1. Multiplication: $x^5 \cdot x^7$ (simplifies to $x^{5+7}=x^{12}$)
2. Division: $\frac{x^{18}}{x^6}$ (simplifies to $x^{18-6}=x^{12}$)
3. Grouping: $(x^4)^3$ (simplifies to $x^{4 \cdot 3}=x^{12}$)
4. Combined: $\frac{(x^3)^5}{x^3}$ (simplifies to $\frac{x^{15}}{x^3}=x^{15-3}=x^{12}$)

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For 10-122: