Question

algebra: concepts and connections - plc
multiplying and dividing rational expressions
which division sentence is related to the product of \\(\frac{a}{3}\\) (\\(\frac{a}{3}\\)) when \\(a \
eq 0\\)?
options:
\\(\frac{a}{3} \div 1 = \frac{a}{3}\\)
\\(\frac{a^2}{9} \div \frac{3}{a} = 1\\)
\\(\frac{a^2}{9} \div \frac{a}{3} = \frac{a}{3}\\)
\\(\frac{a}{3} \div \frac{3}{a} = 1\\)

Explanation:

Step1: Calculate the given product

$$\frac{a}{3} \times \frac{a}{3} = \frac{a \times a}{3 \times 3} = \frac{a^2}{9}$$

Step2: Relate product to division

For $a \times b = c$, the related division is $c \div a = b$. Here, $c=\frac{a^2}{9}$, $a=\frac{a}{3}$, $b=\frac{a}{3}$, so:
$$\frac{a^2}{9} \div \frac{a}{3} = \frac{a}{3}$$

Answer:

$\boldsymbol{\frac{a^2}{9} \div \frac{a}{3} = \frac{a}{3}}$ (the third option)