NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 2 — Figure it Out (Page 22)#
$n^a$ means $n$ multiplied by itself $a$ times. Here $n$ is the base and $a$ is the exponent (or power). For example, $5^4 = 5 \times 5 \times 5 \times 5 = 625$.
1. Express the following in exponential form: (i) 6 × 6 × 6 × 6 (ii) y × y (iii) b × b × b × b (iv) 5 × 5 × 7 × 7 × 7 (v) 2 × 2 × a × a (vi) a × a × a × c × c × c × c × d #
Solution
We count how many times each base is multiplied, and write that count as its exponent:
- (i) $6$ is multiplied 4 times, so $6 \times 6 \times 6 \times 6 = 6^4$
- (ii) $y$ is multiplied 2 times, so $y \times y = y^2$
- (iii) $b$ is multiplied 4 times, so $b \times b \times b \times b = b^4$
- (iv) $5$ twice and $7$ three times, so $5 \times 5 \times 7 \times 7 \times 7 = 5^2 \times 7^3$
- (v) $2$ twice and $a$ twice, so $2 \times 2 \times a \times a = 2^2 \times a^2$
- (vi) $a$ three times, $c$ four times and $d$ once, so $a \times a \times a \times c \times c \times c \times c \times d = a^3 \times c^4 \times d$
2. Express each of the following as a product of powers of their prime factors in exponential form. (i) 648 (ii) 405 (iii) 540 (iv) 3600 #
(i) 648
We keep dividing by the smallest prime that divides the number:
$648 = 2 \times 2 \times 2 \times 81$
$648 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$
$648 = 2^3 \times 3^4$
(ii) 405
$405 = 5 \times 81$
$405 = 3 \times 3 \times 3 \times 3 \times 5$
$405 = 3^4 \times 5$
(iii) 540
$540 = 4 \times 135$
$540 = 2 \times 2 \times 3 \times 3 \times 3 \times 5$
$540 = 2^2 \times 3^3 \times 5$
(iv) 3600
$3600 = 36 \times 100$
$3600 = (2 \times 2 \times 3 \times 3) \times (2 \times 2 \times 5 \times 5)$
$3600 = 2^4 \times 3^2 \times 5^2$
3. Write the numerical value of each of the following: (i) 2 × 10³ (ii) 7² × 2³ (iii) 3 × 4⁴ (iv) (– 3)² × (– 5)² (v) 3² × 10⁴ (vi) (– 2)⁵ × (– 10)⁶ #
(i) 2 × 10³
$2 \times 10^3 = 2 \times 1000$
$2 \times 10^3 = 2000$
(ii) 7² × 2³
$7^2 \times 2^3 = 49 \times 8$
$7^2 \times 2^3 = 392$
(iii) 3 × 4⁴
$3 \times 4^4 = 3 \times 256$
$3 \times 4^4 = 768$
(iv) (– 3)² × (– 5)²
An even power of a negative number is positive:
$(-3)^2 \times (-5)^2 = 9 \times 25$
$(-3)^2 \times (-5)^2 = 225$
(v) 3² × 10⁴
$3^2 \times 10^4 = 9 \times 10000$
$3^2 \times 10^4 = 90000$
(vi) (– 2)⁵ × (– 10)⁶
An odd power of a negative number is negative, and an even power is positive:
$(-2)^5 \times (-10)^6 = (-32) \times 1000000$
$(-2)^5 \times (-10)^6 = -32000000$