Question 15 Marks
Expand $a^3 b^2, a^2 b^3, b^2 a^3, b^3 a^2$ . Are they all same?
Answer
View full question & answer→$a^3 b^2=a^3 \times b^2$
$=(a \times a \times a) \times(b \times b)$
$=a \times a \times a \times b \times b$
$a^2 b^3=a^2 \times b^3$
$=a \times a \times b \times b \times b$
$b^2 a^3=b^2 \times a^3$
$=b \times b \times a \times a \times a$
$b^3 a^2=b^3 \times a^2$
$=b \times b \times b \times a \times a$
Note that in the case of terms $a^3 b^2$ and $a^2 b^3$ the powers of $a$ and $b$ are different. Thus $a^3 b^2$ and $a^2 b^3$ are different. On the other hand, $a^3 b^2$ and $b^2 a^3$ are the same, since the powers of $a$ and $b$ in these two terms are the same. The order of factors does not matter.
Thus, $a^3 b^2=a^3 \times b^2=b^2 \times a^3=b^2 a^3$. Similarly, $a^2 b^3$ and $b^3 a^2$ are the same.
$=(a \times a \times a) \times(b \times b)$
$=a \times a \times a \times b \times b$
$a^2 b^3=a^2 \times b^3$
$=a \times a \times b \times b \times b$
$b^2 a^3=b^2 \times a^3$
$=b \times b \times a \times a \times a$
$b^3 a^2=b^3 \times a^2$
$=b \times b \times b \times a \times a$
Note that in the case of terms $a^3 b^2$ and $a^2 b^3$ the powers of $a$ and $b$ are different. Thus $a^3 b^2$ and $a^2 b^3$ are different. On the other hand, $a^3 b^2$ and $b^2 a^3$ are the same, since the powers of $a$ and $b$ in these two terms are the same. The order of factors does not matter.
Thus, $a^3 b^2=a^3 \times b^2=b^2 \times a^3=b^2 a^3$. Similarly, $a^2 b^3$ and $b^3 a^2$ are the same.