ICSE Class 6 Maths Chapter 07 Powers and Roots

Chapter 7

Powers And Roots

7.1 Exponential Notations (Positive Powers Only)

If number 3 is multiplied to itself 4 times, we get : 3 × 3 × 3 × 3. The product of 3 × 3 × 3 × 3 can be written as \(3^4\). \(3^4\) is called the exponential notation for the product of 3 × 3 × 3 × 3. In the product 3 × 3 × 3 × 3 = \(3^4\), the repeating number 3 is called the base, whereas 4 is called the exponent (or index or power).

Similarly :

(i) in \(2^3\), base = 2 and exponent (power) = 3

(ii) in \((-3)^4\), base = -3 and exponent (power) = 4

(iii) in \(x^{\frac{2}{3}}\), base = x and exponent (index) = \(\frac{2}{3}\) and so on.

In general, \(3^4\) is read as: 3 raised to the power 4 or the fourth power of three.

\((-3)^4 = -3 \times -3 \times -3 \times -3 = 81\) and \(-(3)^4 = -(3 \times 3 \times 3 \times 3) = -81\)

For any non-zero number a, \(a^0 = 1\)

i.e. \(2^0 = 1\), \((-5)^0 = 1\), \(\left(\frac{2}{3}\right)^0 = 1\) and so on.

For any number a, \(a^1 = a\)

i.e. \((-2)^1 = -2\), \(5^1 = 5\), \(\left(\frac{2}{3}\right) = \frac{2}{3}\) and so on.

Example 1

Evaluate : (i) \(3^2 \times 2^4\) (ii) \(3^2 \times 2^0\)

Solution

(i) \(3^2 \times 2^4 = 3 \times 3 \times 2 \times 2 \times 2 \times 2 = 144\)

(ii) \(3^2 \times 2^0 = 3 \times 3 \times 2^0 = 9 \times 1 = 9\)

Example 2

Simplify and express the result in exponential form :

(i) \(\frac{5 \times 2 \times 3^3}{3 \times 3 \times 2 \times 5^3}\) (ii) \(\frac{3^4 \times 2^4}{2^3 \times 3^5}\)

Solution

(i) \(\frac{5 \times 2 \times 3^3}{3 \times 3 \times 2 \times 5^3} = \frac{5 \times 2 \times 3 \times 3 \times 3}{3 \times 3 \times 2 \times 5 \times 5 \times 5} = \frac{3}{5 \times 5} = \frac{3}{5^2}\)

(ii) \(\frac{3^4 \times 2^4}{2^3 \times 3^6} = \frac{3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3} = \frac{2}{3 \times 3} = \frac{2}{3^2}\)

Example 3

Evaluate : \(6a^2b^3\) for a = 3 and b = 2.

Solution

\(6a^2b^3 = 6 \times a \times a \times b \times b \times b = 6 \times 3 \times 3 \times 2 \times 2 \times 2 = 432\)

The following table will make it more clear:

Exercise 7(A)

1. Fill in the blanks :

(i) In expression \(x^y\), base = __________ and exponent = __________

(ii) In expression \((-6)^4\), power = ______________ and base = __________

(iii) If base = 8 and exponent = 5, then expression = ______________

(iv) If base = -2 and power = 10, then expression = ________________

2. Evaluate the following :

(i) \(5^0\) = __________

(ii) \(-5^0\) = __________

(iii) \(-4^3\) = __________

(iv) \((-1)^7\) = __________

(v) If x = 1, then \(15x^7\) = __________

(vi) If y = 3, then \(y^3\) = __________

(vii) If a = -2, then \(4a^3\) = __________

(viii) If \(3^a = 1\), then a = __________

3. Find the value of :

(i) \((-2)^5\)

(ii) \((-10)^3\)

(iii) \(10^4\)

(iv) \(7^4\)

4. Find the value of :

(i) \(\left(\frac{2}{3}\right)^4\)

(ii) \(\left(-\frac{1}{2}\right)^6\)

(iii) \(\left(-\frac{2}{7}\right)^3\)

(iv) \(\left(\frac{3}{5}\right)^4\)

5. Simplify and express the result in exponential notation :

(i) \(\frac{2 \times 2 \times 2 \times 2 \times 3}{3 \times 3 \times 3 \times 2}\)

(ii) \(\frac{3^5 \times 2^2}{3 \times 3 \times 2}\)

(iii) \(\frac{5 \times 5 \times 5^5 \times 3^2}{3 \times 3 \times 5^2}\)

(iv) \(\frac{2^3 \times 3^4 \times 5^2}{5^4 \times 3^2 \times 2}\)

(v) \(\frac{5^6 \times 11^3}{11^5 \times 5 \times 5}\)

6. Evaluate :

(i) \(\frac{(-3)^3 \times (-2)^7}{(-2)^5}\)

(ii) \(8^5 \div 8^2\)

(iii) \((-5)^7 \div 5^4\)

Teacher's Note

Exponential notation helps us write very large numbers compactly, just like we use shorthand in texting - instead of writing a number 1,000,000,000,000, we can write \(10^{12}\).

7. Evaluate :

(i) \(a^3\) for a = 3

(ii) \(4b^3\) for b = 4

(iii) \(2x^3\) for x = 5

(iv) \((3x)^2\) for x = 1

(v) \((4a)^3\) for a = -1

(vi) \(2a^3b^2\) for a = 2 and b = 3

(vii) \(3x^3y^4\) for x = 1 and y = -1

(viii) \(a^b^3\) for a = 2 and b = -3

(ix) \(a^3 + b^3\) for a = 1 and b = 2

(x) \(a^3 + b^3 - 3ab\) for a = 2 and b = 1

8. Express each of the following numbers in the exponential form of its prime factors :

(i) 16

(ii) 81

(iii) 144

(iv) 2700

(v) 3000

(vi) 6075

(vii) 4500

7.2 Properties Of Exponents

Property 1 (Product Law)

\[a^m \times a^n = a^{m+n}\]

For example :

(i) \(a^3 \times a^2 = a^{3+2} = a^5\)

(ii) \(3^7 \times 3^3 = 3^{7+3} = 3^{10}\)

(iii) \((-5)^4 \times (-5)^2 = (-5)^6\)

(iv) \(\left(\frac{3}{4}\right)^6 \times \left(\frac{3}{4}\right)^5 = \left(\frac{3}{4}\right)^{11}\) and so on.

Property 2 (Quotient Law)

\[\frac{a^m}{a^n} = a^{m-n}, \text{ if } m > n\] and \[\frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \text{ if } m < n \text{ and } a \neq 0.\]

For example :

(i) \(\frac{a^5}{a^3} = a^{5-3} = a^2\)

(ii) \(\frac{a^3}{a^5} = \frac{1}{a^{5-3}} = \frac{1}{a^2}\)

(iii) \(\frac{3^7}{3^4} = 3^{7-4} = 3^3\)

(iv) \(\frac{3^4}{3^7} = \frac{1}{3^{7-4}} = \frac{1}{3^3}\) and so on.

For both the properties discussed above, the base must be the same.

Property 3 (Power Law)

\[(a^m)^n = a^{m \times n}\]

For example :

(i) \((a^5)^3 = a^{5 \times 3} = a^{15}\)

(ii) \((3^{-2})^5 = 3^{-2 \times 5} = 3^{-10}\) and so on.

Also, (i) \((a^3 \times b^4)^5 = a^{3 \times 5} \times b^{4 \times 5} = a^{15} \times b^{20}\)

(ii) \(\left(\frac{3^2}{2^4}\right)^3 = \frac{3^{2 \times 3}}{2^{4 \times 3}} = \frac{3^6}{2^{12}}\) and so on.

\((a \times b)^5 = a^5 \times b^5\), but \((a + b)^5 \neq a^5 + b^5\) and \((a - b)^5 \neq a^5 - b^5.\)

Example 4

Using the properties of exponents, evaluate :(i) \(\frac{2^3 \times 2^7}{2^6}\) (ii) \(\frac{(-2)^3 \times 2^7}{2^6}\) (iii) \(\frac{3^8 \times 4^3}{3^6 \times 4^4}\)

Solution

(i) \(\frac{2^3 \times 2^7}{2^6} = \frac{2^{3+7}}{2^6} = \frac{2^{10}}{2^6} = 2^{10-6} = 2^4 = 2 \times 2 \times 2 \times 2 = 16\)

(ii) \(\frac{(-2)^3 \times 2^7}{2^6} = \frac{-2^3 \times 2^7}{2^6}\) [Note: \((-2)^3 = -2 \times -2 \times -2 = -2^3\)] \(= \frac{-2^{10}}{2^6} = -2^{10-6} = -2^4 = -2 \times 2 \times 2 \times 2 = -16\)

(iii) \(\frac{3^8 \times 4^3}{3^6 \times 4^4} = \frac{3^{8-6}}{4^{4-3}} = \frac{3^2}{4^1} = \frac{3 \times 3}{4} = \frac{9}{4} = 2\frac{1}{4}\)

Exercise 7(B)

1. Fill in the blanks :

(i) \(x^a \times x^b\) = ______________

(ii) \(a^3 \times a^8\) = ______________

(iii) \(a^5 \times b^3\) = ______________

(iv) \(2^3 \times 3^2\) = ______________

(v) \(\frac{5^7}{5^4}\) = __________________

(vi) \(\frac{5^3}{5^5}\) = __________________

(vii) \(\frac{8^6}{8^4}\) = __________________

(viii) \((a^3 \times b^4)^2\) = __________

(ix) \(\left(\frac{3^0}{2^2}\right)\) = ______________

(x) \((4^2 \times 5^0)^3\) = __________

2. Using the properties of exponents, evaluate :

(i) \(\frac{3^4 \times 3^3}{3^{15}}\)

(ii) \(\frac{4^6 \times 4^3}{4^5 \times 4^2}\)

(iii) \(\frac{(-2)^6 \times (-2)^5}{(-2)^7}\)

(iv) \(\frac{2^3 \times 5^2}{5^4}\)

(v) \(\frac{5^2 \times 3^6}{3^3}\)

3. Evaluate :

(i) \(\frac{(-3)^3 \times (-2)^4}{(-2)^2}\)

(ii) \(\frac{(2^3 \times 5^4)^2}{2^4 \times 5^5}\)

(iii) \((-5)^4 \div 5^4\)

(iv) \((-3)^7 \div (-3)^6\)

4. Evaluate :

(i) \(2^3 - 3^2 + 4^0\)

(ii) \(5^2 + 2^2 - 3^3 + 8^0\)

5. Evaluate :

(i) \(2x^2\), if x = 3

(ii) \(x^3y^2\), if x = 2 and y = 3

(iii) \(x^2 + y^3\), if x = 1 and y = 2

(iv) \(3x^2y - 2xy^2\), if x = 5 and y = 6

The following example shows the calculation for \(3x^2y - 2xy^2 = 3 \times 5^2 \times 6 - 2 \times 5 \times 6^2 = 3 \times 25 \times 6 - 10 \times 36 = 450 - 360 = 90\)

Teacher's Note

Properties of exponents make calculations with large numbers manageable - scientists use these rules when calculating distances between stars or the size of atoms, making complex computations feasible.

7.3 Squares

When a number is multiplied by itself, the product obtained is called the square of that number.

For example :

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

(ii) Since -2 × -2 = 4, \(\therefore\) 4 is square of -2 and we write : \((-2)^2 = 4\)

(iii) Since \(\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}\), \(\therefore\) \(\frac{4}{9}\) is square of \(\frac{2}{3}\) which is written as \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\)

(iv) Since 0.2 × 0.2 = 0.04 \(\therefore\) 0.04 is square of 0.2 i.e. \((0.2)^2 = 0.04\) and so on.

Whether the number is positive or negative, its square is always positive. e.g. (i) \((3)^2 = 3 \times 3 = 9\)

(ii) \((-3)^2 = -3 \times -3 = 9\)

(iii) \((-5)^2 = -5 \times -5 = 25\) and so on

More examples :

(i) Square of 0 = \(0^2 = 0\)

(ii) Square of 5 = \(5^2 = 5 \times 5 = 25\)

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

(iv) Square of \(2\frac{3}{7} = \left(2\frac{3}{7}\right)^2 = \left(\frac{17}{7}\right)^2 = \frac{17}{7} \times \frac{17}{7} = \frac{289}{49} = 5\frac{44}{49}\)

(v) Square of \(-3\frac{1}{2} = \left(-3\frac{1}{2}\right)^2 = \left(-\frac{7}{2}\right)^2 = \left(-\frac{7}{2}\right) \times \left(-\frac{7}{2}\right) = \frac{49}{4} = 12\frac{1}{4}\)

(vi) Square of -2.3 = \((-2.3)^2 = -2.3 \times -2.3 = 5.29\) and so on.

Exercise 7(C)

1. Find the squares of first five natural numbers.

2. Find the squares of first six even natural numbers.

3. Find the squares of first four odd natural numbers.

4. Find the squares of first five prime numbers.

5. Find the squares of :

(i) 9

(ii) \(\frac{2}{5}\)

(iii) \(1\frac{2}{7}\)

(iv) \(2\frac{3}{4}\)

6. Find the squares of :

(i) -3

(ii) \(-\frac{2}{3}\)

(iii) \(-1\frac{2}{5}\)

(iv) \(-2\frac{1}{4}\)

7. Find the squares of :

(i) 2.5

(ii) 0.6

(iii) 0.23

(iv) 0.02

(v) -1.6

(vi) -0.8

Teacher's Note

Understanding squares helps us in everyday situations like calculating the area of a square room or piece of land, which is essential for interior design, construction, and real estate pricing.

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