Question

Annabelle and Navene are multiplying (4^27^5)(4^37^2).

Annabelle's Work:

(4^27^5)(4^37^2) = 4^2 + ^37^5 + ^2 = 4^57^7


Navene's Work:

(4^27^5)(4^37^2) = 4^2⋅^37^5⋅^2 = 4^67^10


Is either of them correct? Explain your reasoning. (Say Who and Why Please) (Will Give Brainliest)

Answer 1

Correct question.

Annabelle and Navene are multiplying $4^{2}\cdot7^{5}$ and $4^{3}\cdot 7^{2}$.

Annabelle's Work:

$(4^{2}\cdot7^{5})\cdot (4^{3}\cdot 7^{2})=4^{2+3}\cdot 7^{5+2}=4^5\cdot 7^7$

Navene's Work:

$(4^{2}\cdot7^{5})\cdot (4^{3}\cdot 7^{2})=4^{2\cdot3}\cdot 7^{5\cdot2}=4^6\cdot 7^{10}$

Reasoning.

$\rightarrow a^m\times a^n=(a\times a\times...\times a)\times (a\times a\times...\times a)$

Here, the first bracket contains '$m$' $a$'s and the other contains '$n$'

How many $a$'s are there? There are a total of '$m+n$' $a$'s.

$\rightarrow a^{m}\times a^{n}=a^{m+n}$

Annabelle added the power and Navene multiplied the power. So, Annabelle is correct.

Learn more.

Product rule $a^m\times a^n=a^{m+n}$

Division rule $a^{m}\div a^{n}=a^{m-n}$

Power rule $(a^m)^{n}=a^{mn}$

Power rule $a^{m^{n}}=a^{(m^{n})}$

Root rule $\sqrt[m]{a} =a^{\frac{1}{m} }$

→ For even power root, the base should be non-negative.

Zero power rule $a^{0}=1\;(a\neq 0)$

Negative power rule $\dfrac{1}{a^m} =a^{-m}\;(a\neq 0)$