# DAV Class 7 Maths Chapter 4 Worksheet 3 Solutions

The DAV Class 7 Maths Solutions and DAV Class 7 Maths Chapter 4 Worksheet 3 Solutions of Application of Percentage offer comprehensive answers to textbook questions.

## DAV Class 7 Maths Ch 4 WS 3 Solutions

Question 1.
Which of the following are true?
(i) $$\left(+\frac{3}{4}\right)^3 \times\left(\frac{-3}{4}\right)^3=\left(\frac{-3}{4}\right)^6$$
$$\left(+\frac{3}{4}\right)^3 \times\left(\frac{-3}{4}\right)^3=\left(\frac{-3}{4}\right)^6$$
Here bases are not same
∴ it is false.

(ii) $$\left(\frac{4}{7}\right)^5 \times\left(\frac{4}{7}\right)^3=\left(\frac{4}{7}\right)^8$$
$$\left(\frac{4}{7}\right)^5 \times\left(\frac{4}{7}\right)^3=\left(\frac{4}{7}\right)^8$$
⇒ $$\left(\frac{4}{7}\right)^{5+3}=\left(\frac{4}{7}\right)^8$$
⇒ $$\left(\frac{4}{7}\right)^8=\left(\frac{4}{7}\right)^8$$
Hence it is true.

(iii) $$\left(\frac{-1}{2}\right)^4 \div\left(\frac{-1}{2}\right)^3=\left(\frac{-1}{2}\right)$$
$$\left(\frac{-1}{2}\right)^4 \div\left(\frac{-1}{2}\right)^3=\left(\frac{-1}{2}\right)$$
⇒ $$\left(\frac{-1}{2}\right)^{4-3}=\left(\frac{-1}{2}\right)$$
⇒ $$\left(\frac{-1}{2}\right)=\left(\frac{-1}{2}\right)$$
Hence it is true.

(iv) $$\left(\frac{6}{7}\right)^6 \div\left(\frac{6}{7}\right)^0=\left(\frac{6}{7}\right)^0$$

⇒ $$\left(\frac{6}{7}\right)^{6-0}=\left(\frac{6}{7}\right)^0$$
⇒ $$\left(\frac{6}{7}\right)^6 \neq\left(\frac{6}{7}\right)^0$$
Hence it is false.

Questions 2.
Fill in the blanks in each of the following so as to make the statements true:
(i) $$\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right)^3=\left(\frac{-2}{3}\right)^{\cdots}$$
$$\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right)^3=\left(\frac{-2}{3}\right)$$
⇒ $$\left(\frac{-2}{3}\right)^{1+3}=\left(\frac{-2}{3}\right)^4$$
∴ (…………..) = 4

(ii) $$\left(\frac{4}{5}\right)^{\cdots} \times\left(\frac{4}{5}\right)^9=\left(\frac{4}{5}\right)^{11}$$
$$\left(\frac{4}{5}\right)^{\cdots} \times\left(\frac{4}{5}\right)^9=\left(\frac{4}{5}\right)^{11}$$
⇒ $$\left(\frac{4}{5}\right)^{\cdots}=\left(\frac{4}{5}\right)^{11} \div\left(\frac{4}{9}\right)^9$$
⇒ $$\left(\frac{4}{5}\right)^{\cdots}=\left(\frac{4}{5}\right)^{11-9}$$
⇒ $$\left(\frac{4}{5}\right)^2=\left(\frac{4}{5}\right)^2$$
∴ (…………..) = 2

(iii) $$\left(\frac{-3}{7}\right)^{15} \div\left(\frac{-3}{7}\right)^{\cdots}=\left(\frac{-3}{7}\right)$$
$$\left(\frac{-3}{7}\right)^{15} \div\left(\frac{-3}{7}\right)^{\cdots}=\left(\frac{-3}{7}\right)$$
⇒ $$\left(\frac{-3}{7}\right)^{15} \div\left(\frac{-2}{3}\right)=\left(\frac{-3}{7}\right)^{\cdots}$$
⇒ $$\left(\frac{-3}{7}\right)^{15-1}=\left(\frac{-3}{7}\right)^{\cdots}$$
⇒ $$\left(\frac{-3}{7}\right)^{14}=\left(\frac{-3}{7}\right)^{-1}$$
∴ (…………..) = 14

(iv) $$\left(\frac{-1}{10}\right)^{\cdots} \div\left(\frac{-1}{10}\right)^8=\left(\frac{-1}{10}\right)^8$$
$$\left(\frac{-1}{10}\right)^{\cdots} \div\left(\frac{-1}{10}\right)^8=\left(\frac{-1}{10}\right)^8$$
⇒ $$\left(\frac{-1}{10}\right)^{\cdots}=\left(\frac{-1}{10}\right)^8 \times\left(\frac{-1}{10}\right)^8$$
⇒ $$\left(\frac{-1}{10}\right)^{\cdots}=\left(\frac{-1}{10}\right)^{8+8}$$
⇒ $$\left(\frac{-1}{10}\right)^{\cdots}=\left(\frac{-1}{10}\right)^{16}$$
∴ (…………..) = 16

(v) $$\left(\frac{2}{9}\right)^6 \div\left(\frac{2}{9}\right)^0=\left(\frac{2}{9}\right)^{\cdots}$$
$$\left(\frac{2}{9}\right)^6 \div\left(\frac{2}{9}\right)^0=\left(\frac{2}{9}\right)^{\cdots}$$
⇒ $$\left(\frac{2}{9}\right)^{6-0}=\left(\frac{2}{9}\right)^{\cdots}$$
⇒ $$\left(\frac{2}{9}\right)^6=\left(\frac{2}{9}\right)^3$$
∴ (…………..) = 6

(vi) $$\left(\frac{-12}{13}\right)^2 \times\left(\frac{-12}{13}\right)^{\cdots}=\left(\frac{-12}{13}\right)^5$$
$$\left(\frac{-12}{13}\right)^2 \times\left(\frac{-12}{13}\right)^{\cdots}=\left(\frac{-12}{13}\right)^5$$
⇒ $$\left(\frac{-12}{13}\right)^{\cdots}=\left(\frac{-12}{13}\right)^5 \div\left(\frac{-12}{13}\right)^2$$
⇒ $$\left(\frac{-12}{13}\right)=\left(\frac{-12}{13}\right)^{5-2}$$
⇒ $$\left(\frac{-12}{13}\right)^{\cdots}=\left(\frac{-12}{13}\right)^3$$
∴ (…………..) = 3

Question 3.
Simplify and express the result in exponential form:
(i) $$\left(\frac{-3}{7}\right)^3 \times\left(\frac{-3}{7}\right)^4$$
$$\left(\frac{-3}{7}\right)^3 \times\left(\frac{-3}{7}\right)^4$$
= $$\left(\frac{-3}{4}\right)^{3+4}=\left(\frac{-3}{4}\right)^7$$

(ii) $$\left(\frac{11}{12}\right)^{15} \times\left(\frac{11}{12}\right)^5 \times\left(\frac{11}{12}\right)^{10}$$
$$\left(\frac{11}{12}\right)^{15} \times\left(\frac{11}{12}\right)^5 \times\left(\frac{11}{12}\right)^{10}$$
⇒ $$\left(\frac{11}{12}\right)^{15+5+10}=\left(\frac{11}{12}\right)^{30}$$

(iii) $$\left(\frac{-9}{11}\right)^9 \div\left(\frac{-9}{11}\right)^7$$
$$\left(\frac{-9}{11}\right)^9 \div\left(\frac{-9}{11}\right)^7$$
= $$\left(\frac{-9}{11}\right)^{9-7}=\left(\frac{-9}{11}\right)^2$$

(iv) $$\left(\frac{1}{4}\right)^8 \div\left(\frac{1}{4}\right)^6$$
$$\left(\frac{1}{4}\right)^8 \div\left(\frac{1}{4}\right)^6$$
= $$\left(\frac{1}{4}\right)^{8-6}=\left(\frac{1}{4}\right)^2$$

(v) $$\left(\frac{15}{19}\right)^{35} \div\left(\frac{15}{19}\right)^{24}$$
$$\left(\frac{15}{19}\right)^{35} \div\left(\frac{15}{19}\right)^{24}$$
= $$\left(\frac{15}{19}\right)^{35-24}=\left(\frac{15}{19}\right)^{11}$$

Question 4.
Simplify and express the results as a rational number.
(i) $$\left(\frac{5}{7}\right)^4 \div\left(\frac{5}{7}\right)^2$$
$$\left(\frac{5}{7}\right)^4 \div\left(\frac{5}{7}\right)^2=\left(\frac{5}{7}\right)^{4-2}$$
= $$\left(\frac{5}{7}\right)^2=\frac{5}{7} \times \frac{5}{7}$$
= $$\frac{25}{49}$$

(ii) $$\left(\frac{-2}{3}\right)^2 \times\left(\frac{-2}{3}\right)^3$$
$$\left(\frac{-2}{3}\right)^2 \times\left(\frac{-2}{3}\right)^3=\left(\frac{-2}{3}\right)^{2+3}$$
= $$\left(\frac{-2}{3}\right)^5=\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right)$$
= $$\frac{-32}{243}$$

(iii) $$\left(\frac{3}{4}\right)^3 \times\left(\frac{3}{4}\right)^2$$
$$\left(\frac{3}{4}\right)^3 \times\left(\frac{3}{4}\right)^2=\left(\frac{3}{4}\right)^{3+2}$$
= $$\left(\frac{3}{4}\right)^5$$
= $$\frac{243}{1024}$$

(iv) $$\left(\frac{-3}{5}\right)^6 \div\left(\frac{-3}{5}\right)^3$$
$$\left(\frac{-3}{5}\right)^6 \div\left(\frac{-3}{5}\right)^2=\left(\frac{-3}{5}\right)^{6-2}$$
= $$\left(\frac{-3}{5}\right)^4$$
= $$\frac{81}{625}$$

(v) $$\left(\frac{-1}{10}\right)^4 \times\left(\frac{-1}{10}\right)^2$$
$$\left(\frac{-1}{10}\right)^4 \times\left(\frac{-1}{10}\right)^2=\left(\frac{-1}{10}\right)^{4+2}$$
= $$\left(\frac{-1}{10}\right)^6$$
= $$\frac{1}{1000000}$$

Question 5.
Evaluate:
(i) $$\left(\frac{3}{4}\right)^3 \times\left(\frac{2}{3}\right)^2$$
$$\left(\frac{3}{4}\right)^3 \times\left(\frac{2}{3}\right)^2$$

= $$\frac{3}{16}$$

(ii) $$\left[\left(\frac{1}{3}\right)^6 \div\left(\frac{1}{3}\right)^5\right] \div \frac{1}{3}$$
$$\left[\left(\frac{1}{3}\right)^6 \div\left(\frac{1}{3}\right)^5\right] \div\left(\frac{1}{3}\right)$$
= $$\left(\frac{1}{3}\right)^{6-5} \div \frac{1}{3}=\frac{1}{3} \div \frac{1}{3}$$
= 1

(iii) (24 × 25) ÷ 28
= 24+5 ÷ 28
= 29 ÷ 28
= 29-8
= 2

(iv) [42 – 32] ÷ $$\left(\frac{1}{7}\right)^2$$
= (16 – 9) ÷ $$\left(\frac{1}{7}\right)^2$$
= (7) ÷ (7)-2
= 71+2
= 73
= 343.