Question
Expand $a^3 b^2, a^2 b^3, b^2 a^3, b^3 a^2$ . Are they all same?
✓
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.
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Let us try to find a few more pairs of numbers from their sums and differences:
(a) Sum = 27, Difference = 9
(b) Sum = 4, Difference = 12
(c) Sum = 0, Difference = 10
(d) Sum = 0, Difference = -10
(e) Sum = -7, Difference = -1
(f) Sum = -7, Difference = -13
(a) Sum = 27, Difference = 9
(b) Sum = 4, Difference = 12
(c) Sum = 0, Difference = 10
(d) Sum = 0, Difference = -10
(e) Sum = -7, Difference = -1
(f) Sum = -7, Difference = -13
Construct 4 pairs of parallel lines in different orientations.
Case-Based Question :
Rita is organising a school fair. She is in charge of budgeting for various stalls. She notes the following expenses :
→ The cost of each food stall is ₹ x.
→ The cost of each game stall is ₹ y.
→ She plans to have 4 food stalls and 3 game stalls.
There is a fixed decoration cost of ₹ 500.
Based on the above information, answer the following questions.
(a) Write an expression for the total cost to organise the school fair.
(b) What does the term 4x represent?
(c) If the cost of each food stall is ₹200 and each game stall is ₹ 150, what will be the total cost?
Rita is organising a school fair. She is in charge of budgeting for various stalls. She notes the following expenses :
→ The cost of each food stall is ₹ x.
→ The cost of each game stall is ₹ y.
→ She plans to have 4 food stalls and 3 game stalls.
There is a fixed decoration cost of ₹ 500.
Based on the above information, answer the following questions.
(a) Write an expression for the total cost to organise the school fair.
(b) What does the term 4x represent?
(c) If the cost of each food stall is ₹200 and each game stall is ₹ 150, what will be the total cost?
Write each of the following rational numbers with positive denomi-natirs: $\frac{5}{-8}, \frac{15}{-28},\frac{-17}{-13}.$
→In Fig. the radius of quarter circular plot taken is $2m$ and radius of the flower bed is $2m$. Find the area of the remaining field.
→Find the value of $x^2+y^2+2 z^2+2 x y+3 y z+4 z x$ when $x=2, y=-3$ and z = 4.
→If milk is available at ₹ $17 \frac{3}{4}$ per litre, find the cost of $7 \frac{2}{5}$ litres of milk.
→The length and breadth of a playground are $62m$ $60\ cm$ and $25m$ $40\ cm$ respectively. Find the cost of turfing it at $Rs. 2.50$ per square-metre. How long will a man take to go three times round the field, if he walks at the rate of $2$ metres per-second?
→The boundary of shaded region in the given figure consists of three semi-circular areas, the small one being equal. If diameter of large one with centre $O$ is 30 cm , then calculate
(i) the length of boundary of shaded region.
(ii) the area of shaded region.
→(i) the length of boundary of shaded region.
(ii) the area of shaded region.
The drawings below $Fig.$ show angles formed by the goalposts at different positions of a football player. The greater the angle, the better chance the player has of scoring a goal. For example, the player has a better chance of scoring a goal from Position $A$ than from Position $B.$
$i.$
$ii.$
$iii.$
In Parts $(a)$ and $(b)$ given below it may help to trace the diagrams and draw and measure angles.
$a.$ Seven football players are practicing their kicks. They are lined up in a straight line in front of the goalpost $[Fig. (ii)].$ Which player has the best $($thegreatest$)$ kicking angle?
$b.$ Now the players are lined up as shown in $Fig. (iii)$. Which player has the best kicking angle?
$c.$ Estimate atleast two situations such that the angles formed by different positions of two players are complement to each other.
→$i.$
$ii.$
$iii.$
In Parts $(a)$ and $(b)$ given below it may help to trace the diagrams and draw and measure angles.
$a.$ Seven football players are practicing their kicks. They are lined up in a straight line in front of the goalpost $[Fig. (ii)].$ Which player has the best $($thegreatest$)$ kicking angle?
$b.$ Now the players are lined up as shown in $Fig. (iii)$. Which player has the best kicking angle?
$c.$ Estimate atleast two situations such that the angles formed by different positions of two players are complement to each other.