Q1. Write each of the following products in exponential form:
$$ (i) \quad a \times a \times a \times a \times \ldots \ldots 15 \text { times} $$ $$ (ii) \quad 8 \times b \times b \times b \times a \times a \times a \times a $$ $$ (iii) \quad 5 \times a \times a \times a \times b \times b \times c \times c \times c $$$$ (iv) \quad 7 \times a \times a \times a \ldots 8 \text { times} \times b \times b \times b \times \ldots 5 \text { times} $$ $$ (v) \quad 4 \times a \times a \times \ldots 5 \text { times} \times b \times b \times \ldots 12 \text { times} \times c \times c \ldots 15 \text { times} $$Sol. Respective exponential forms are
$$ (i) \quad a^{15} $$ $$ (ii) \quad 8 b^{3} a^{4}=8 a^{4} b^{3} $$ $$ (iii) \quad 5 a^{3} b^{2} c^{3} $$ $$ (iv) \quad 7 a^{8} b^{5} $$ $$ (v) \quad 4 a^{5} b^{12} c^{15} $$Q2. Write each of the following in the product form:
$$ (i) \quad a^{2} b^{5} $$ $$ (ii) \quad 8 x^{3} $$ $$ (iii) \quad 7 a^{3} b^{4} $$ $$ (iv) \quad 15 a^{9} b^{8} c^{6} $$ $$ (v) \quad 30 x^{4} y^{4} z^{5} $$ $$ (vi) \quad 43 p^{10} q^{5} r^{15} $$ $$ (vii) \quad 17 p^{12} q^{20} $$$$ Sol. (i) \quad a \times a \times b \times b \times b \times b \times b $$ $$ (ii) \quad 2 \times 2 \times 2 \times x \times x \times x $$ $$ (iii) \quad 7 \times a \times a \times a \times b \times b \times b \times b $$$$ (iv) \quad 3 \times 5 \times a \times a \times a \ldots 9 \text { times} \times b \times b \times b \ldots 8 \text { times} \times c \times c \times c \ldots 6 \text { times} $$$$ (v) \quad 2 \times 3 \times 5 \times x \times x \times x \times x \times y \times y \times y \times y \times z \times z \times z \times z \times z $$$$ (vi) \quad 43 \times p \times p \times p \ldots 10 \text { times} \times q \times q \times q \ldots 5 \text { times} \times r \times r \times r \ldots 15 \text { times} $$$$ (vii) \quad 17 \times p \times p \times p \times \ldots 12 \text { times} \times q \times q \times q \times \ldots 20 \text { times} $$Q3. Write down each of the following in exponential form:
$$ (i) \quad 4 a^{3} \times 6 a b^{2} \times c^{2} $$ $$ (ii) \quad 5 x y \times 3 x^{2} y \times 7 y^{2} $$ $$ (iii) \quad a^{3} \times 3 a b^{2} \times 2 a^{2} b^{2} $$$$ Sol. (i) \quad 4 a^{3} \times 6 a b^{2} \times c^{2} $$ $$ =4 \times 6 \times a^{3} \times a \times b^{2} \times c^{2} $$ $$ =24 a^{4} b^{2} c^{2} $$$$ (ii) \quad 5 x y \times 3 x^{2} y \times 7 y^{2} $$ $$ =5 \times 3 \times 7 \times x \times x^{2} \times y \times y \times y^{2} $$ $$ =105 x^{3} y^{4} $$$$ (iii) \quad a^{3} \times 3 a b^{2} \times 2 a^{2} b^{2} $$ $$ = 3 \times 2 \times a^{3} \times a \times a^{2} \times b^{2} \times b^{2} $$ $$ =6 a^{6} b^{4} $$Q4. The number of bacteria in a culture is x now. It becomes square of itself after one week. What will be its number after two weeks?
Sol. Current number of bacteria in the culture = x
Given that bacteria becomes square of itself in a week.
$$ \therefore \text {Number of bacteria in the culture after one week } =x^{2} $$ $$ \Rightarrow \text {Number of bacteria in the culture after two weeks } $$ $$ = (\text {No. of bacteria after 1 week})^{2} $$ $$ =(x^{2})^{2}=x^{4} $$Q5. 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.
$$ \text {Sol. Given that area of rectangle } = \text {length} \times \text {breadth} $$Breadth of the given rectangle = x cm
$$ \therefore \text {Length of the rectangle } =\frac{2}{3} \times x \text { cm} $$ $$ \Rightarrow \text {Area of the rectangle } =\frac{2}{3}x \times x=\frac{2}{3} x^{2} \text { cm}^{2} $$$$ \text {Q6. If there are x rows of chairs and each row contains } x^{2} \text { chairs. Determine the total number of chairs.} $$$$ \text {Sol. Given that there are x rows of chairs and each row contains } x^{2} \text { chairs.} $$ $$ \text { Total number of chairs } = \text {Number of rows } \times \text {Number of chairs in each row } $$ $$ =x \times x^{2} =x^{3} $$
$$ (i) \quad a \times a \times a \times a \times \ldots \ldots 15 \text { times} $$ $$ (ii) \quad 8 \times b \times b \times b \times a \times a \times a \times a $$ $$ (iii) \quad 5 \times a \times a \times a \times b \times b \times c \times c \times c $$$$ (iv) \quad 7 \times a \times a \times a \ldots 8 \text { times} \times b \times b \times b \times \ldots 5 \text { times} $$ $$ (v) \quad 4 \times a \times a \times \ldots 5 \text { times} \times b \times b \times \ldots 12 \text { times} \times c \times c \ldots 15 \text { times} $$Sol. Respective exponential forms are
$$ (i) \quad a^{15} $$ $$ (ii) \quad 8 b^{3} a^{4}=8 a^{4} b^{3} $$ $$ (iii) \quad 5 a^{3} b^{2} c^{3} $$ $$ (iv) \quad 7 a^{8} b^{5} $$ $$ (v) \quad 4 a^{5} b^{12} c^{15} $$Q2. Write each of the following in the product form:
$$ (i) \quad a^{2} b^{5} $$ $$ (ii) \quad 8 x^{3} $$ $$ (iii) \quad 7 a^{3} b^{4} $$ $$ (iv) \quad 15 a^{9} b^{8} c^{6} $$ $$ (v) \quad 30 x^{4} y^{4} z^{5} $$ $$ (vi) \quad 43 p^{10} q^{5} r^{15} $$ $$ (vii) \quad 17 p^{12} q^{20} $$$$ Sol. (i) \quad a \times a \times b \times b \times b \times b \times b $$ $$ (ii) \quad 2 \times 2 \times 2 \times x \times x \times x $$ $$ (iii) \quad 7 \times a \times a \times a \times b \times b \times b \times b $$$$ (iv) \quad 3 \times 5 \times a \times a \times a \ldots 9 \text { times} \times b \times b \times b \ldots 8 \text { times} \times c \times c \times c \ldots 6 \text { times} $$$$ (v) \quad 2 \times 3 \times 5 \times x \times x \times x \times x \times y \times y \times y \times y \times z \times z \times z \times z \times z $$$$ (vi) \quad 43 \times p \times p \times p \ldots 10 \text { times} \times q \times q \times q \ldots 5 \text { times} \times r \times r \times r \ldots 15 \text { times} $$$$ (vii) \quad 17 \times p \times p \times p \times \ldots 12 \text { times} \times q \times q \times q \times \ldots 20 \text { times} $$Q3. Write down each of the following in exponential form:
$$ (i) \quad 4 a^{3} \times 6 a b^{2} \times c^{2} $$ $$ (ii) \quad 5 x y \times 3 x^{2} y \times 7 y^{2} $$ $$ (iii) \quad a^{3} \times 3 a b^{2} \times 2 a^{2} b^{2} $$$$ Sol. (i) \quad 4 a^{3} \times 6 a b^{2} \times c^{2} $$ $$ =4 \times 6 \times a^{3} \times a \times b^{2} \times c^{2} $$ $$ =24 a^{4} b^{2} c^{2} $$$$ (ii) \quad 5 x y \times 3 x^{2} y \times 7 y^{2} $$ $$ =5 \times 3 \times 7 \times x \times x^{2} \times y \times y \times y^{2} $$ $$ =105 x^{3} y^{4} $$$$ (iii) \quad a^{3} \times 3 a b^{2} \times 2 a^{2} b^{2} $$ $$ = 3 \times 2 \times a^{3} \times a \times a^{2} \times b^{2} \times b^{2} $$ $$ =6 a^{6} b^{4} $$Q4. The number of bacteria in a culture is x now. It becomes square of itself after one week. What will be its number after two weeks?
Sol. Current number of bacteria in the culture = x
Given that bacteria becomes square of itself in a week.
$$ \therefore \text {Number of bacteria in the culture after one week } =x^{2} $$ $$ \Rightarrow \text {Number of bacteria in the culture after two weeks } $$ $$ = (\text {No. of bacteria after 1 week})^{2} $$ $$ =(x^{2})^{2}=x^{4} $$Q5. 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.
$$ \text {Sol. Given that area of rectangle } = \text {length} \times \text {breadth} $$Breadth of the given rectangle = x cm
$$ \therefore \text {Length of the rectangle } =\frac{2}{3} \times x \text { cm} $$ $$ \Rightarrow \text {Area of the rectangle } =\frac{2}{3}x \times x=\frac{2}{3} x^{2} \text { cm}^{2} $$$$ \text {Q6. If there are x rows of chairs and each row contains } x^{2} \text { chairs. Determine the total number of chairs.} $$$$ \text {Sol. Given that there are x rows of chairs and each row contains } x^{2} \text { chairs.} $$ $$ \text { Total number of chairs } = \text {Number of rows } \times \text {Number of chairs in each row } $$ $$ =x \times x^{2} =x^{3} $$