Question

1. 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. (cpm 3 - 7) a. \\( \frac { x _ { 150 } } { x _ { 50 } } \

ightarrow x _ { 3 } \\) b. \\( y _ { 20 } \cdot y _ { 41 } \
ightarrow y _ { 61 } \\) c. \\( ( 2 m _ { 15 } ^ { 2 } n ) ^ { 3 } \
ightarrow 2 m _ { 45 } ^ { 6 } n \\)

Explanation:

Step1: Check rule for division of exponents

When dividing exponents with the same base \(a^m\div a^n=a^{m - n}\). For \(a.\frac{x^{150}}{x^{50}}\), using the rule \(x^{150-50}=x^{100}
eq x^{3}\). So Gerardo's result for \(a\) is incorrect.

Step2: Check rule for multiplication of exponents

When multiplying exponents with the same base \(a^m\times a^n=a^{m + n}\). For \(b.y^{20}\cdot y^{41}\), using the rule \(y^{20 + 41}=y^{61}\). So Gerardo's result for \(b\) is correct.

Step3: Check rule for power - of - a - power

When raising a power to a power \((a^m)^n=a^{m\times n}\). For \(c.(2m^{15}n^{2})^3\), using the rule \(2^3\times m^{15\times3}\times n^{2\times3}=8m^{45}n^{6}
eq2m^{45}n^{6}\). So Gerardo's result for \(c\) is incorrect.

Answer:

a. Incorrect, because \(\frac{x^{150}}{x^{50}}=x^{100}
eq x^{3}\).
b. Correct, because \(y^{20}\cdot y^{41}=y^{20 + 41}=y^{61}\).
c. Incorrect, because \((2m^{15}n^{2})^3 = 8m^{45}n^{6}
eq2m^{45}n^{6}\).