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#### Page No 8.11:

#### Question 1:

Write each of the following products in exponential form:

(i) *a* × *a* × *a* × *a* × ..... 15 times

(ii) 8 × *b* ×* b* ×* b* × *a* × *a* × *a* × *a*

(iii) 5 × *a *× *a *× *a* × *b* × *b* × *c *× *c* × *c*

(iv) 7 × *a *× *a* × *a*.... 8 times × *b* × *b* × *b* ×.... 5 times

(v) 4 × *a* × *a* ×..... 5 times × *b *×* b* ×....12 times × *c* ×* c*.... 15 times

#### Answer:

$\left(i\right){a}^{15}\phantom{\rule{0ex}{0ex}}\left(ii\right)8{b}^{3}{a}^{4}=8{a}^{4}{b}^{3}\phantom{\rule{0ex}{0ex}}\left(iii\right)5{a}^{3}{b}^{2}{c}^{3}\phantom{\rule{0ex}{0ex}}\left(iv\right)7{a}^{8}{b}^{5}\phantom{\rule{0ex}{0ex}}\left(v\right)4{a}^{5}{b}^{12}{c}^{15}$

#### Page No 8.12:

#### Question 2:

Write each of the following in the product form:

(i) *a*^{2}*b*^{5}

(ii) 8*x*^{3}

(iii) 7*a*^{3}*b*^{4}

(iv) 15*a*^{9}*b*^{8}*c*^{6}

(v) 30*x*^{4}*y*^{4}*z*^{5}

(vi) 43*p*^{10}*q*^{5}*r*^{15}

(vii) 17*p*^{12}*q*^{20}

#### Answer:

$\left(\mathrm{i}\right)a\times a\times b\times b\times b\times b\times b\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right)2\times 2\times 2\times x\times x\times x\phantom{\rule{0ex}{0ex}}\left(\mathrm{iii}\right)7\times a\times a\times a\times b\times b\times b\times b\phantom{\rule{0ex}{0ex}}\left(\mathrm{iv}\right)3\times 5\times a\times a\times a\times a\times a\times a\times a\times a\times a\times b\times b\times b\times b\times b\times b\times b\times b\times c\times c\times c\times c\times c\times c$

$\left(\mathrm{v}\right)2\times 3\times 5\times x\times x\times x\times x\times y\times y\times y\times y\times z\times z\times z\times z\times z\phantom{\rule{0ex}{0ex}}\left(\mathrm{vi}\right)43\times p\times p\times p\times p....10\mathrm{times}\times q\times q....5\mathrm{times}\times r\times r\times r....15\mathrm{times}\phantom{\rule{0ex}{0ex}}\left(\mathrm{vii}\right)17\times p\times p\times ....12\mathrm{times}\times q\times q\times ....20\mathrm{times}$

#### Page No 8.12:

#### Question 3:

Write down each of the following in exponential form:

(i) 4*a*^{3} × 6*ab*^{2} × *c*^{2}

(ii) 5*xy* × 3*x*^{2}*y* × 7*y*^{2}

(iii) *a*^{3} × 3*ab*^{2} × 2*a*^{2}*b*^{2}

#### Answer:

$\left(\mathrm{i}\right)4\times 6\times {a}^{3}\times a\times {b}^{2}\times {c}^{2}\phantom{\rule{0ex}{0ex}}=24{a}^{4}{b}^{2}{c}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right)5\times 3\times 7\times x\times {x}^{2}\times y\times y\times {y}^{2}\phantom{\rule{0ex}{0ex}}=105{x}^{3}{y}^{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{iii}\right)3\times 2\times {a}^{3}\times a\times {a}^{2}\times {b}^{2}\times {b}^{2}\phantom{\rule{0ex}{0ex}}=6{a}^{6}{b}^{4}$

#### Page No 8.12:

#### Question 4:

The number of bacteria in a culture is *x* now. If becomes square of itself after one week. What will be its number after two weeks?

#### Answer:

Present number of bacteria in a culture = *x*

Number of bacteria in the culture after one week = ${x}^{2}$

Number of bacteria in the culture after two weeks = $({x}^{2}{)}^{2}={x}^{4}$

#### Page No 8.12:

#### Question 5:

The area of a rectangle is given by the product of its length and breadth. The length of a rectangle is two-third of its breadth. Find its area if its breadth is *x* cm.

#### Answer:

Breadth of the given rectangle = *x *cm

Length of the rectangle = $\frac{2}{3}x$ cm

$\therefore $ Area of the rectangle = $\frac{2}{3}x\times x=\frac{2}{3}{x}^{2}$ cm^{2}

#### Page No 8.12:

#### Question 6:

If there are *x* rows of chairs and each row contains *x*^{2} chairs. Determine the total number of chairs.

#### Answer:

Total number of chairs = Number of rows $\times $ Number of chairs in each row

$=x\times {x}^{2}\phantom{\rule{0ex}{0ex}}={x}^{3}$

#### Page No 8.13:

#### Question 1:

5 more than twice a number *x* is written as

(a) 5 + *x* + 2

(b) 2*x* + 5

(c) 2*x* − 5

(d) 5*x* + 2

#### Answer:

(b) 2*x* + 5

#### Page No 8.13:

#### Question 2:

The quotient of *x* by 2 is added to 5 is writen as

(a) $\frac{x}{2}+5$

(b) $\frac{2}{x}+5$

(c) $\frac{x+2}{5}$

(d) $\frac{x}{2+5}$

#### Answer:

(a) $\frac{x}{2}+5$

#### Page No 8.13:

#### Question 3:

The quotient of *x* by 3 is multiplied by *y* is written as

(a) $\frac{x}{3y}$

(b) $\frac{3x}{y}$

(c) $\frac{3y}{x}$

(d) $\frac{xy}{3}$

#### Answer:

(d) $\frac{xy}{3}$

$\frac{x}{3}\times y=\frac{xy}{3}$

#### Page No 8.13:

#### Question 4:

9 taken away from the sum of *x* and *y* is

(a) x + y − 9

(b) 9 − (x+y)

(c) $\frac{x+y}{9}$

(d) $\frac{9}{x+y}$

#### Answer:

(a) x + y − 9

#### Page No 8.13:

#### Question 5:

The quotient of *x* by *y* added ot the product of *x *and *y* is written as

(a) $\frac{x}{y}+xy$

(b) $\frac{y}{x}+xy$

(c) $\frac{xy+x}{y}$

(d) $\frac{xy+y}{x}$

#### Answer:

(a) $\frac{x}{y}+xy$

#### Page No 8.13:

#### Question 6:

*a*^{2}*b*^{3} × 2*ab*^{2} is equal to

(a) 2*a*^{3}*b*^{4}

(b) 2*a*^{3}*b*^{5}

(c) 2*ab*

(d) *a*^{3}*b*^{5}

#### Answer:

(b) 2*a*^{3}*b*^{5}

${a}^{2}{b}^{3}\times 2a{b}^{2}\phantom{\rule{0ex}{0ex}}=2{a}^{2}\times a\times {b}^{3}\times {b}^{2}\phantom{\rule{0ex}{0ex}}=2{a}^{3}{b}^{5}$

#### Page No 8.13:

#### Question 7:

4*a*^{2}*b*^{3} × 3*ab*^{2} × 5*a*^{3}*b* is equal to

(a) 60*a*^{3}*b*^{5}

(b) 60*a*^{6}*b*^{5}

(c) 60*a*^{6}*b*^{6}

(d) *a*^{6}*b*^{6}

#### Answer:

(c) 60*a*^{6}*b*^{6}

$4{a}^{2}{b}^{3}\times 3a{b}^{2}\times 5{a}^{3}b\phantom{\rule{0ex}{0ex}}=4\times 3\times 5\times {a}^{2}\times a\times {a}^{3}\times {b}^{3}\times {b}^{2}\times b\phantom{\rule{0ex}{0ex}}=60{a}^{6}{b}^{6}$

#### Page No 8.13:

#### Question 8:

If 2*x*^{2}*y* and 3*xy*^{2} denote the length and breadth of a rectangle, the its area is

(a) 6*xy*

(b) 6*x*^{2}*y*^{2}

(c) 6*x*^{3}*y*^{3}

(d) *x*^{3}*y*^{3}

#### Answer:

(c) 6*x*^{3}*y*^{3}

Area of the rectangle = Length $\times $ Breadth

= $2{x}^{2}y\times 3x{y}^{2}\phantom{\rule{0ex}{0ex}}=6{x}^{3}{y}^{3}$

#### Page No 8.13:

#### Question 9:

In a room there are* **x*^{2} rows of chairs and each two contains 2*x*^{2} chairs. The total number of chairs in the room is

(a) 2*x*^{3}

(b) 2*x*^{4}

(c) *x*^{4}

(d) $\frac{{x}^{4}}{2}$

#### Answer:

(b) 2*x*^{4}

Total number of chairs in the room = Number of rows $\times $ Number of chairs in each row

= *x*^{2} $\times $ 2*x*^{2} = 2*x*^{4}

#### Page No 8.13:

#### Question 10:

*a*^{3} × 2*a*^{2}*b* × 3*ab*^{5} is equal to

(a) *a*^{6}*b*^{6}

(b) 23*a*^{6}*b*^{6}

(c) 6*a*^{6}*b*^{6}

(d) None of these

#### Answer:

(c) 6*a*^{6}*b*^{6}

*a*^{3} × 2*a*^{2}*b* × 3*ab*^{5}

^{$=2\times 3{a}^{3}\times {a}^{2}\times a\times b\times {b}^{5}\phantom{\rule{0ex}{0ex}}=6{a}^{(3+2+1)}{b}^{(1+5)}\phantom{\rule{0ex}{0ex}}=6{a}^{6}{b}^{6}$}

#### Page No 8.7:

#### Question 1:

Write the following using numbers, literals and sings of basic operations. State what each letter represents:

(i) The diameter of a circle is twice its radius.

(ii) The area of a rectangle is the product of its length and breadth.

(iii) The selling price equals the sum of the cost price and the profit.

(iv) The total amount equals the sum of the principal and the interest.

(v) The perimeter of a rectangle is two times the sum of its length and breadth.

(vi) The perimeter of a square is four times its side.

#### Answer:

(i) Let *r* and *d* be the radius and diameter of the circle, respectively.

$\therefore $ *d* = 2*r*

(ii) Let *l* and *b* be the length and breadth of the rectangle, respectively.

$\therefore $ Area of rectangle = *lb*

(iii) Let *s*, *c* and *p* be the selling price, cost price and profit, respectively.

$\therefore $* s* = *c *+ *p*

(iv) Let *T*, *p* and *i* be the total amount, principal and interest, respectively.

$\therefore $ *T* = *p* + *i*

(v) Let *l* and *b* be the length and breadth of the rectangle, respectively.

$\therefore $ Perimeter of rectangle = 2(*l *+ *b*)

(vi) Let *a* be the side of the square.

$\therefore $ Perimeter of the square = 4*a*

#### Page No 8.7:

#### Question 2:

Write the following using numbers, literals and sings of basic operations:

(i) The sum of 6 and *x*.

(ii) 3 more than a number *y.*

(iii) One-third of a number *x*.

(iv) One-half of the sum of number *x* and *y*.

(v) Number* y* less than a number 7.

(vi) 7 taken away from *x*.

(vii) 2 less than the quotient of *x* by *y.*

(viii) 4 time *x* taken away from one-third of *y.*

(ix) Quotient of *x* by 3 is multiplied by *y*.

#### Answer:

(i) The sum of 6 and *x* is 6 + *x.*

(ii) 3 more than a number *y *means *y* + 3*.*

(iii) One-third of a number *x* is* $\frac{x}{3}$.*

(iv) One-half of the sum of numbers *x* and *y* is $\frac{(x+y)}{2}$.

(v) Number* y* less than a number 7 means 7 $-$ *y.*

(vi) 7 taken away from *x* means *x *$-$ 7.

(vii) 2 less than the quotient of *x* by *y *is $\frac{x}{y}-2$.

(viii) 4 times *x* taken away from one-third of *y *is* *$\frac{y}{3}-4x$.

(ix) Quotient of *x* by 3 is multiplied by *y *means:

$\frac{x}{3}\times y=\frac{xy}{3}$

#### Page No 8.8:

#### Question 3:

Think of a number. Multiply it by 5. Add 6 to the result. Subtract *y* from this result. What is the result?

#### Answer:

Let the number be *x*.

On multiplying the number by 5, we get:

5*x*

Further, adding 6 to 5*x*, we get:

5*x* + 6

Finally, on subtracting *y* from 5*x* + 6, we get:

5*x* + 6 $-$ *y*

#### Page No 8.8:

#### Question 4:

The number of rooms on the ground floor of a building is 12 less than the twice of the number of rooms on first floor. If the first floor has *x *rooms, how many rooms does the ground floor has?

#### Answer:

Let the number of rooms on the ground floor be *y*.

It is given that the number of rooms on the first floor is *x*; therefore, we have:

*y = *2* $\times $ x $-$ *12

* = *2*x $-$ *12

Thus, the number of rooms on the ground floor is 2*x $-$ *12.

#### Page No 8.8:

#### Question 5:

Binny spends Rs *a* daily and saves Rs *b* per week. What is her income for two weeks?

#### Answer:

It is given that Binny spends Rs. *a *in one day.

$\therefore $ Money spent by him in one week = 7 $\times $ *a = *7*a*

It is further given that he saves Rs. *b *in one week; therefor we have:

Total income in one week = Total expenditure in one week + Total saving in one week

= 7*a** + b*

$\therefore $ Binny's total income in 2 weeks = 2 $\times $ (7*a** + b*) = Rs. (14*a** + *2*b*)

#### Page No 8.8:

#### Question 6:

Rahul scores 80 marks in English and *x* marks in Hindi. What is his total score in the two subject?

#### Answer:

$\mathrm{Marks}\mathrm{obtained}\mathrm{in}\mathrm{English}=80\phantom{\rule{0ex}{0ex}}\mathrm{Marks}\mathrm{obtained}\mathrm{in}\mathrm{Hindi}=x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Total}\mathrm{marks}\mathrm{obtained}=80+x\phantom{\rule{0ex}{0ex}}$

#### Page No 8.8:

#### Question 7:

Rohit covers *x* centimeters in one step. How much distance does he cover in *y* steps?

#### Answer:

It is given that Rohit covers *x* cm in one step.

$\therefore $ Distance covered by him in *y* steps = *x $\times $ y = xy *cm

#### Page No 8.8:

#### Question 8:

One apple weighs 75 grams and one orange weighs 40 grams. Determine the weight of *x* apples and *y* oranges.

#### Answer:

$\mathrm{Weight}\mathrm{of}\mathrm{an}\mathrm{apple}=75\mathrm{grams}\phantom{\rule{0ex}{0ex}}\mathrm{Weight}\mathrm{of}\mathrm{an}\mathrm{orange}=40\mathrm{grams}\phantom{\rule{0ex}{0ex}}\mathrm{Weight}\mathrm{of}x\mathrm{apples}=75\times x=75x\mathrm{grams}\phantom{\rule{0ex}{0ex}}\mathrm{Weight}\mathrm{of}\mathrm{y}\mathrm{oranges}=40\times \mathrm{y}=40y\mathrm{grams}\phantom{\rule{0ex}{0ex}}\mathrm{Total}\mathrm{weight}\mathrm{of}x\mathrm{apples}\mathrm{and}y\mathrm{oranges}=(75x+40y)\mathrm{grams}$

#### Page No 8.8:

#### Question 9:

One pencil costs Rs 2 and one fountain pen costs Rs 15. What is the cost of *x* pencils and *y* fountain pens?

#### Answer:

Cost of one pencil = Rs. 2

Cost of *x *pencils = Rs. 2*x*

Cost of one fountain pen = Rs. 15

Cost of *y* fountain pens = Rs. 15*y*

Total cost of *x* pencils and *y* fountain pens = Rs. (2*x* + 15*y*)

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