ICSE Class 7 Maths Chapter 02 Power and Roots

Chapter 2: Power And Roots

Elementary Treatment

1. Power (Exponent)

(i) We know \(5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^7\), where 5 is called the base and 7 is called the power or exponent or index of 5.

(ii) \(-2 \times -2 \times -2 \times \ldots\) 20 times \(= (-2)^{20}\), in which 20 is the power (index or exponent) of base -2.

2. Properties Of Exponents

First Property (Product Law)

For any non-zero integer a:

\(a^m \times a^n = a^{m+n}\), \(a^m \times a^n \times a^p = a^{m+n+p}\) and so on.

For example:

(i) \(3^4 \times 3^6 = 3^{4+6} = 3^{10}\)

(ii) \(5^8 \times 5^5 = 5^{8+5} = 5^{13}\)

(iii) \((-2)^6 \times (-2)^7 = (-2)^{6+7} = (-2)^{13}\) and so on.

Thus, the product of two or more numbers in exponent form (all having the same base) is a number with the same base (as that of the given numbers) and whose exponent is equal to the sum of the exponents of the numbers multiplied together.

Second Property (Quotient Law)

For any non-zero integer a:

\(\frac{a^m}{a^n} = a^{m-n}\), if m > n and \(\frac{a^m}{a^n} = \frac{1}{a^{n-m}}\), if n > m

For example:

(i) \(\frac{3^8}{3^6} = 3^{8-6} = 3^2\) and \(\frac{3^6}{3^8} = \frac{1}{3^{8-6}} = \frac{1}{3^2}\)

(ii) \(\frac{(-5)^4}{(-5)^{10}} = \frac{1}{(-5)^{10-4}} = \frac{1}{(-5)^6}\) and \(\frac{(-5)^{10}}{(-5)^4} = (-5)^{10-4} = (-5)^6\) and so on.

Third Property (Power Law)

For any non-zero integer a:

\((a^m)^n = a^{mn} = a^{nm}\)

For example:

(i) \((2^3)^4 = 2^{3 \times 4} = 2^{12}\)

(ii) \((8^5)^2 = 8^{5 \times 2} = 8^{10}\)

(iii) \((7^4)^5 = 7^{20}\), \([(-7)^4]^5 = (-7)^{20}\) and so on.

Also:

(i) \(a^0 = 1 \Rightarrow 2^0 = 1, 3^{20} = 1, (-5)^0 = 1\) and so on.

(ii) \(a^1 = 1 \Rightarrow 2^1 = 2, 3^1 = 32, (-5)^1 = -5\) and so on.

(iii) \(a^{-m} = \frac{1}{a^m} \Rightarrow 2^{-3} = \frac{1}{2^3}, 3^{-2} = \frac{1}{3^2}, (-5)^{-7} = \frac{1}{(-5)^7}\) and so on.

(iv) \(\frac{1}{a^{-m}} = a^m \Rightarrow \frac{1}{2^{-5}} = 2^5, \frac{1}{5^{-8}} = 5^8, \frac{1}{(-3)^{-6}} = (-3)^6\) and so on.

(i) If m is even, \((-a)^m = a^m\)

i.e., \((-2)^4 = 2^4, (-8)^{10} = 8^{10}, (-5)^6 = 5^6\) and so on

(ii) If m is odd, \((-a)^m = -a^m\)

i.e., \((-2)^5 = -2^5, (-8)^9 = -8^9, (-5)^7 = -5^7\) and so on

(iii) Any non-zero number raised to the power zero = 1

i.e., \(3^0 = 1, (-3)^0 = 1\), but \(-3^0 \neq 1\). Infact, \(-3^0\) means \(-( 3^0) = -1\).

Similarly: \(15^0 = 1, (-15)^0 = 1\), but \(-15^0 = -1\), \(28^0 = 1, (-28)^0 = 1\), but \(-28^0 = -1\) and so on.

Also,

(iv) \(-2^4 = -2 \times 2 \times 2 \times 2 = -16\), whereas \((-2)^4 = -2 \times -2 \times -2 \times -2 = 16\)

or, as exponent 4 is even; \((-2)^4 = 2^4 = 2 \times 2 \times 2 \times 2 = 16\)

(v) \((-2)^5 = -2 \times -2 \times -2 \times -2 \times -2 = -32\)

or, \((-2)^5 = -2^5 = -2 \times 2 \times 2 \times 2 \times 2 = -32\)

Example 1

Evaluate: (i) \(3^3 \times 3^5 \div 3^6\) (ii) \((-2)^7 \times (-2)^6 \div (-2)^{10}\)

Solution

(i) \(3^3 \times 3^5 \div 3^6 = \frac{3^3 \times 3^5}{3^6}\) [Applying BODMAS]

\(= \frac{3^8}{3^6} = 3^{8-6} = 3^2 = 3 \times 3 = 9\)

(ii) \((-2)^7 \times (-2)^6 \div (-2)^{10} = \frac{(-2)^7 \times (-2)^6}{(-2)^{10}} = \frac{(-2)^{13}}{(-2)^{10}} = (-2)^{13-10}\)

\(= (-2)^3 = -2 \times -2 \times -2 = -8\)

Example 2

Evaluate: (i) \(3^3 \div 3^5 \times 3^6\) (ii) \((-2)^7 \div (-2)^6 \times (-2)^2\)

Solution

(i) \(3^3 \div 3^5 \times 3^6 = \frac{3^3}{3^5} \times 3^6\) [Applying BODMAS]

\(= \frac{3^3 \times 3^6}{3^5} = \frac{3^9}{3^5} = 3^{9-5} = 3^4 = 3 \times 3 \times 3 \times 3 = 81\)

(ii) \((-2)^7 \div (-2)^6 \times (-2)^2\)

\(= \frac{(-2)^7}{(-2)^6} \times (-2)^2 = (-2)^1 \times (-2)^2 = (-2)^3 = -2^3 = -8\)

Example 3

Evaluate: (i) \(2^6 - 3^0 \times 2^5\) (ii) \(2 \times 3^4 - (-2)^3 + (-4)^2\)

Solution

(i) Since, \(2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64, 3^0 = 1\) and \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\)

\(\therefore 2^6 - 3^0 \times 2^5 = 64 - 1 \times 32\)

\(= 64 - 32 = 32\)

(ii) Since, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

\((-2)^3 = -2 \times -2 \times -2 = -8\)

or, \((-2)^3 = -2^3 = -2 \times 2 \times 2 = -8\)

and, \((-4)^2 = -4 \times -4 = 16\)

\(\therefore 2 \times 3^4 - (-2)^3 + (-4)^2 = 2 \times 81 - (-8) + 16 = 162 + 8 + 16 = 186\)

Exercise 2(A)

1. Fill In The Blanks

(i) In \((5)^{-7}\); 5 is called - and -7 is called -.

(ii) In \(8^3\); 3 is called - and 8 is called -.

(iii) \(2^5 \times 2^7 = -\), \(2^5 \times 2^7 \times 2^{-6} = -\) and \(2^5 \times 2^7 \times 2^{-6} \times 2^3 = -\).

(iv) \(4^8 \div 4^5 = -\), \(4^8 \times 4^3 \div 4^5 = -\) and \(4^8 \div 4^3 \times 4^5 = -\)

(v) If \(3^x = 1; x = -\).

(vi) For every value of x, \(1^x = -\).

(vii) \((2^3)^2 = - = - = -\).

(viii) \((2^5)^0 = -, (7^3)^0 = -\) and \((-8)^0 = -\).

(ix) \(5^0 = -\) and \(-5^0 = -\).

(x) \((-8)^0 = -\) and \(-8^0 = -\).

2. Evaluate

(i) \(4^8 \times 4^{-6}\) (ii) \(3^7 \times 3^{-5} \times 3\) (iii) \(5^5 \times 5^4 \times 5^{-6}\)

(iv) \(7^4 \times 7^0 \times 7^{-3}\) (v) \(8^3 \times 8^2 \times 8^{-5}\)

3. Evaluate

(i) \(2^6 \times 2^4 \div 2^8\) (ii) \(2^6 \div 2^7 \times 2^3\) (iii) \(3^{10} \times 3^{15} \times 3^6\)

(iv) \(5^{10} \times 5^{12} \div 5^{19}\) (v) \((-7)^3 \div (-7)^5 \times (-7)^4\)

4. Evaluate

(i) \((5^4)^2 \times 5^{-6}\) (ii) \((3^{-2})^4 \times (3)^9\) (iii) \((7^{-3})^4 \times (7^{-2})^{-7}\)

(iv) \((4^{-4})^6 \times (4^3)^2 \times (4^{-2})^2\)

5. Evaluate

(i) \(2^8 - 5^0 \times 2^6\) (ii) \(3 \times 4^3 - 4 \times 5^2 + (2)^3 \times 8^0\)

(iii) \(7 \times 3^2 + 5 \times 4^3 \times 6^0 - 6 \times 2^6\)

Teacher's Note

Exponents help us express very large and very small numbers compactly, like the distance between planets or the size of atoms in a single manageable notation.

Section 2.2: Square Of A Number

If a number be multiplied by itself, the product obtained is called the square of the number.

Thus, (i) square of 4 = 4 × 4 = 16

(ii) square of -5 = -5 × -5 = 25 and so on.

Since, (i) square of 4 is 16; we write: \((4)^2 = 16\)

(ii) square of -5 is 25; we write: \((-5)^2 = 25\) and so on.

Whether the number is positive or negative; its square is always positive.

e.g., (i) \((-5)^2 = -5 \times -5 = 25\) and \((5)^2 = 5 \times 5 = 25\)

(ii) \((7)^2 = 7 \times 7 = 49\) and \((-7)^2 = -7 \times -7 = 49\) and so on.

But; \(-5^2 = -5 \times 5 = -25\); as the square is for 5 only.

\(-7^2 = -7 \times 7 = -49\); as the square is for 7 only.

Section 2.3: Cube Of A Number

If a number be multiplied by itself three times, the product obtained is called the cube of the number.

Thus, (i) cube of 3 = 3 × 3 × 3 = 27, i.e., \((3)^3 = 27\)

(ii) cube of -4 = -4 × -4 × -4 = -64, i.e., \((-4)^3 = -64\)

(iii) cube of \(\frac{3}{5} = \frac{3}{5} \times \frac{3}{5} \times \frac{3}{5} = \frac{27}{125}\), i.e., \(\left(\frac{3}{5}\right)^3 = \frac{27}{125}\) and so on.

Cube of a positive number is positive and cube of a negative number is negative.

e.g. \((6)^3 = 6 \times 6 \times 6 = 216\) and

\((-6)^3 = -6 \times -6 \times -6 = -216\).

Consider The Following Table

Teacher's Note

Squares and cubes appear everywhere - from calculating the area of a room (length squared) to determining the volume of a storage box (length cubed).

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