# DAV Class 7 Maths Chapter 4 Worksheet 1 Solutions

The DAV Class 7 Maths Solutions and DAV Class 7 Maths Chapter 4 Worksheet 1 Solutions of Application of Percentage offer comprehensive answers to textbook questions.

## DAV Class 7 Maths Ch 4 WS 1 Solutions

Question 1.
Write the base and exponent in each of the following:
(i) $$\left(\frac{-1}{3}\right)^3$$
$$\left(\frac{-1}{3}\right)^3$$, base = $$\frac{-1}{3}$$ and exponent = 3

(ii) $$\left(\frac{-4}{7}\right)^6$$
$$\left(\frac{-4}{7}\right)^6$$, base = $$\frac{-4}{7}$$ and exponent = 6

(iii) $$\left(\frac{2}{9}\right)^5$$
$$\left(\frac{2}{9}\right)^5$$, base = $$\frac{2}{9}$$ and exponent = 5

(iv) $$\left(\frac{15}{19}\right)^3$$
$$\left(\frac{15}{19}\right)^3$$, base = $$\frac{15}{19}$$ and exponent = 3

(v) (-15)4
(-15)4, base = -15 and exponent = 4

(vi) $$\frac{-2}{3}$$
$$\frac{-2}{3}=\left(\frac{-2}{3}\right)^1$$, Here base = $$\frac{-2}{3}$$ and exponent = 1

Question 2.
Express the following in exponential form:
(i) $$\frac{5}{6} \times \frac{5}{6}$$
$$\frac{5}{6} \times \frac{5}{6}=\left(\frac{5}{6}\right)^2$$

(ii) $$\frac{9}{2} \times \frac{9}{2} \times \frac{9}{2} \times \frac{9}{2}$$
$$\frac{9}{2} \times \frac{9}{2} \times \frac{9}{2} \times \frac{9}{2}=\left(\frac{9}{2}\right)^4$$

(iii) $$\left(\frac{-7}{8}\right) \times\left(\frac{-7}{8}\right) \times\left(\frac{-7}{8}\right)$$
$$\left(\frac{-7}{8}\right) \times\left(\frac{-7}{8}\right) \times\left(\frac{-7}{8}\right)=\left(\frac{-7}{8}\right)^3$$

(iv) $$\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right)$$
$$\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right) \times\left(\frac{-1}{2}\right)=\left(\frac{-1}{2}\right)^5$$

(v) 1.8 × 1.8 × 1.8 × 1.8 × 1.8 × 1.8 × 1.8
1.8 × 1.8 × 1.8 × 1.8 × 1.8 × 1.8 × 1.8 = (1.8)7

(vi) $$\frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12}$$
$$\frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12}=\left(\frac{11}{12}\right)^6$$

Question 3.
Express the following as rational numbers in the form
(i) $$\left(\frac{5}{6}\right)^3$$
$$\left(\frac{5}{6}\right)^3=\frac{5}{6} \times \frac{5}{6} \times \frac{5}{6}$$
= $$\frac{125}{216}$$

(ii) $$\left(\frac{-12}{13}\right)^2$$
$$\left(\frac{4}{9}\right)^4=\frac{4}{9} \times \frac{4}{9} \times \frac{4}{9} \times \frac{4}{9}$$
= $$\frac{256}{6561}$$

(iii) $$\left(\frac{4}{9}\right)^4$$
$$\left(-\frac{1}{2}\right)^5=\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right)$$
= $$-\frac{1}{32}$$

(iv) $$\left(-\frac{1}{2}\right)^5$$
$$\left(-\frac{1}{2}\right)^5=\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right)$$
= $$\left(-\frac{1}{2}\right)^5=\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right) \times\left(-\frac{1}{2}\right)$$

(v) $$\left(\frac{1}{4}\right)^4$$
$$\left(\frac{1}{4}\right)^4=\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}$$
= $$\frac{1}{256}$$

(vi) $$\left(\frac{3}{5}\right)^3$$
$$\left(\frac{3}{5}\right)^3=\frac{3}{5} \times \frac{3}{5} \times \frac{3}{5}$$
= $$\frac{27}{125}$$

Question 4.
Express the following as powers of rational numbers:
(i) $$\frac{81}{625}$$
$$\frac{81}{625}=\frac{3 \times 3 \times 3 \times 3}{5 \times 5 \times 5 \times 5}$$
= $$\frac{3}{5} \times \frac{3}{5} \times \frac{3}{5} \times \frac{3}{5}=\left(\frac{3}{5}\right)^4$$

(ii) $$\frac{-8}{125}$$
$$\frac{-8}{125}=\frac{-2 \times 2 \times 2}{5 \times 5 \times 5}$$
= $$\left(\frac{-2}{5}\right) \times\left(\frac{-2}{5}\right) \times\left(\frac{-2}{5}\right)=\left(\frac{-2}{5}\right)^3$$

(iii) $$\frac{-343}{512}$$
$$\frac{-343}{512}=\frac{-7 \times 7 \times 7}{8 \times 8 \times 8}$$
= $$\left(\frac{-7}{8}\right) \times\left(\frac{-7}{8}\right) \times\left(\frac{-7}{8}\right)=\left(\frac{-7}{8}\right)^3$$

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

(v) $$\frac{-1}{216}$$
$$\frac{-1}{216}=\frac{-1 \times 1 \times 1}{6 \times 6 \times 6}$$
= $$\left(\frac{-1}{6}\right) \times\left(\frac{-1}{6}\right) \times\left(\frac{-1}{6}\right)=\left(\frac{-1}{6}\right)^3$$

(vi) $$\frac{729}{1000}$$
$$\frac{729}{1000}=\frac{9 \times 9 \times 9}{10 \times 10 \times 10}$$
= $$\frac{9}{10} \times \frac{9}{10} \times \frac{9}{10}=\left(\frac{9}{10}\right)^3$$.

### DAV Class 8 Maths Chapter 4 Worksheet 1 Notes

Exponent is the power raised to any number.
For example:
(i) ab, here, b is the exponent and a is called the base.
(ii) (- 5)1/3, Here $$\frac{1}{3}$$ is the power of exponent and – 5 is the base.
It is read as – 5 raised to the power 4
$$\left(\frac{3}{4}\right)^4=\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4}$$

Laws of exponents:
(i) If m is the Power of any rational number $$\frac{a}{b}$$, b ≠ 0 then
$$\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}$$ and $$\left(\frac{a}{b}\right)^0$$ = 1
(ii) Reciprocal of $$\left(\frac{a}{b}\right)^m$$ is $$\left(\frac{b}{a}\right)^m$$ if m is a positive integer.
(iii) xm xn = xm + n, x ≠ 0
(iv) xm ÷ xn = xm – n, x ≠ 0
(v) xm ÷ xn = $$\frac{1}{x^{n-m}}$$ if m < n and x ≠ 0
If x is non-zero rational number, then x0 = 1.
(vi) x– m = $$\frac{1}{x^m}$$ i.e., x is the reciprocal of x
(vii) $$\left(\frac{p}{q}\right)^{-m}=\left(\frac{q}{p}\right)^m$$ if $$\frac{p}{q}$$ ≠ 0
(viii) am x bm = (ab)m
(ix) am ÷ bm = $$\left(\frac{a}{b}\right)^m$$
(x) [ab]c = abc

Every number, large or small can be expressed in the form k x l0 where k is a terminating decimal satisfying 1 ≤ k < 10 and n an integer (positive for large numbers and negative for small numbers).

Example 1:
Express the following as rational numbers.

(i) $$\left(\frac{3}{4}\right)^3$$
$$\left(\frac{3}{4}\right)^3$$
= $$\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4}$$
= $$\frac{27}{64}$$

(ii) $$\left(\frac{-5}{7}\right)^2$$
$$\left(\frac{-5}{7}\right)^2$$
= $$\frac{-5}{7} \times \frac{-5}{7}$$
= $$\frac{25}{49}$$

(iii) $$\left(\frac{-4}{5}\right)^{-2}$$
$$\left(\frac{-4}{5}\right)^{-2}$$
= $$\left(\frac{-5}{4}\right)^2$$
= $$\frac{-5}{4} \times \frac{-5}{4}$$
= $$\frac{25}{16}$$

(iv) $$\left(\frac{2}{3}\right)^0$$
$$\left(\frac{2}{3}\right)^0$$ = 1

Example 2:
Express the following rational numbers in exponential form.

(i) $$\frac{125}{64}$$
$$\frac{125}{64}$$ = $$\frac{5 \times 5 \times 5}{4 \times 4 \times 4}$$
= $$\frac{5^3}{4^3}$$
= $$\left(\frac{5}{4}\right)^3$$

(ii) $$\frac{625}{1296}$$
$$\frac{625}{1296}$$
= $$\frac{5 \times 5 \times 5 \times 5}{6 \times 6 \times 6 \times 6}$$
= $$\frac{5^4}{6^4}$$
= $$\left(\frac{5}{4}\right)^4$$

(iii) $$\frac{343}{512}$$
$$\frac{343}{512}$$
= $$\frac{7 \times 7 \times 7}{8 \times 8 \times 8}$$
= $$\frac{7^3}{8^3}$$
= $$\left(\frac{7}{8}\right)^3$$

(iv) $$\frac{64}{729}$$
$$\frac{64}{729}$$
= $$\frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3 \times 3}$$
= $$\frac{2^6}{3^6}$$
= $$\left(\frac{2}{3}\right)^6$$

Example 3:
Find the reciprocal of the following.

(i) $$\left(\frac{3}{5}\right)^{-3}$$
Reciprocal of $$\left(\frac{3}{5}\right)^{-3}$$ = $$\frac{1}{\left(\frac{3}{5}\right)^{-3}}$$
= $$\left(\frac{3}{5}\right)^3$$
= $$\left(\frac{5}{3}\right)^{-3}$$

(ii) $$\left(\frac{-5}{7}\right)^2$$
Reciprocal of $$\left(\frac{-5}{7}\right)^2$$ = $$\frac{1}{\left(\frac{-5}{7}\right)^2}$$
= $$\left(\frac{-7}{5}\right)^2$$

(iii) (0)3
Reciprocal of (0)3 is not defined.

(iv) $$\left(\frac{1}{5}\right)^{-4}$$
Reciprocal of $$\left(\frac{1}{5}\right)^{-4}$$ = $$\frac{1}{\left(\frac{1}{5}\right)^4}$$
= (5)– 4

Example 4:
Simplify the following:
(i) $$\left(\frac{4}{5}\right)^6 \div\left(\frac{4}{5}\right)^4$$
$$\left(\frac{4}{5}\right)^6 \div\left(\frac{4}{5}\right)^4$$ = $$\left(\frac{4}{5}\right)^{6-4}$$
= $$\left(\frac{4}{5}\right)^2$$
= $$\frac{4 \times 4}{5 \times 5}$$
= $$\frac{16}{25}$$

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

Example 5:
Simplify the following:

(i) $$\left[\left(-\frac{1}{5}\right)^2\right]^3$$
$$\left[\left(-\frac{1}{5}\right)^2\right]^3$$ = $$\left(\frac{-1}{5}\right)^{2 \times 3}$$
= $$\left(\frac{-1}{5}\right)^6$$
= $$\frac{1}{15625}$$

(ii) $$\left[\left(\frac{2}{3}\right)^8\right]^{\frac{1}{4}}$$
$$\left[\left(\frac{2}{3}\right)^8\right]^{\frac{1}{4}}$$ = $$\left(\frac{2}{3}\right)^{8 \times \frac{1}{4}}$$
= $$\left(\frac{2}{3}\right)^2$$
= $$\frac{4}{9}$$

Example 6:
Find the value of x, if $$\left[\left(\frac{3}{4}\right)^3\right]^2=\left(\frac{3}{4}\right)^{3 x}$$
$$\left[\left(\frac{3}{4}\right)^3\right]^2=\left(\frac{3}{4}\right)^{3 x}$$
= $$\left(\frac{3}{4}\right)^{3 \times 2}=\left(\frac{3}{4}\right)^{3 x}$$
= $$\left(\frac{3}{4}\right)^6=\left(\frac{3}{4}\right)^{3 x}$$
Since bases are same, their exponents must be same.
∴ 3x = 6
⇒ x = 2.

Example 7:
By what number should (- 3)3 be multiplied so that the product may be $$\frac{1}{27}$$.
Let the required number be x
∴ x × (- 3)– 3 = $$\frac{1}{27}$$
x = $$\frac{1}{27}$$ ÷ (- 3)– 3
x = $$\frac{1}{27} \div \frac{1}{(-3)^3}$$
x = $$\frac{1}{27}$$ × (- 3)– 3
x = $$\frac{1}{27}$$ × (- 27)
∴ x = – 1

Example 8:
Find, if the following are true.
(i) 23 × 33 = (2 × 3)3
(ii) (- 2)4 ÷ (- 3)4 = $$\left(\frac{-2}{3}\right)^4$$
(iii) (- 3)2 + (- 2)2 = (- 3 – 2)2
(i) 23 × 33 = 8 × 27 = 216
(2 × 3)3 = (6)3
= 6 × 6 × 6
= 216

(ii) (- 2)4 ÷ (- 3)4 = $$\left(\frac{-2}{-3}\right)^4$$
= $$\frac{2^4}{3^4}$$
= $$\frac{16}{81}$$
$$\left(\frac{-2}{3}\right)^4$$ = $$\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right)$$
= $$\frac{16}{81}$$
Hence (- 2)4 ÷ (- 3)4 = $$\left(\frac{-2}{-3}\right)^4$$

(iii) (- 3)2 + (- 2)2 = (- 3 – 2)2
(- 3)2 + (- 2)2 = 9 + 4 = 13
(- 3 – 2)2 = (- 5)2 = 25
Hence (- 3)2 + (- 2)2 ≠ (- 3 – 2)2