ICSE Class 8 Maths Chapter 13 Exponents

Chapter 13: Exponents

13.1 Review

Exponent

If x is a real number and n is a natural number, we know:

x × x × x × x ....... n times = xⁿ

where xⁿ is called an exponential expression with base x and exponent (or index, or power) n.

xⁿ is read as 'x raised to the power n' or simply 'x to the power n'.

Laws of Exponents

1. Product Law: a^m × aⁿ = a^{m + n}

e.g. 3⁷ × 3⁴ = 3^{7 + 4} = 3¹¹, x⁸ × x⁵ = x^{8 + 5} = x¹³ and so on.

2. Quotient Law: \[\frac{a^m}{a^n} = a^{m-n} \text{ if } m > n\]

\[= \frac{1}{a^{n-m}} \text{ if } n > m\]

e.g. \[\frac{3^7}{3^4} = 3^{7-4} = 3^3, \frac{x^5}{x^8} = \frac{1}{x^{8-5}} = \frac{1}{x^3}\] and so on.

3. Power law: (a^m)ⁿ = a^mn

e.g. (3⁷)⁴ = 3^{7 × 4} = 3²⁸, (x⁸)⁵ = x⁴⁰ and so on.

Test Yourself

1. 2 × 2 × 2 × 2 .............. 15 times = .............. and is read as: ............................................

2. -5 × -5 × -5 × .............. 12 times = .............. and is read as: ......................................

3. a⁵ × a⁷ = ............., a⁵ × a^-7 = ............., a^-5 × a⁷ = ............ and a^-5 × a^-7 = ............

4. \[\frac{a^8}{a^2} = ............., \frac{a^5}{a^8} = ............., \frac{a^5}{a^{-8}} = ............. \text{ and } \frac{a^8}{a^{-5}} = .............\]

5. (a⁵)⁸ = ............., (a⁸)⁵ = ............., (a^-8)⁵ = .............. and (a^-8)^-5 = ............

6. 3¹⁵ × 3⁶ × 3^-10 = ............., 5⁴ × 5^-7 × 5⁶ = ............ and 7² × 7⁸ × 7^-6 = ............

7. \[\frac{2^5 \times 2^4}{2^9} = ............., \frac{4^6 \times 4^{-3}}{4^2} = ............ \text{ and } \frac{8^5 \times 8^4}{8^{-3}} = ............\]

Teacher's Note

Understanding exponents helps us express very large numbers like the distance to stars or very small numbers like the size of atoms in a simple, compact form.

13.2 More About Exponents

1. (a × b)ⁿ = aⁿ × bⁿ

e.g. (a⁵ × b^-3)⁴ = (a⁵)⁴ × (b^-3)⁴ = a²⁰ × b^-12 and (3⁴ × 5^-3)^-2 = 3^-8 × 5⁶

2. \[\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\]

e.g. \[\left(\frac{a^{-3}}{b^4}\right)^6 = \frac{(a^{-3})^6}{(b^4)^6} = \frac{a^{-18}}{b^{24}} \text{ and } \left(\frac{5^7}{3^{-4}}\right)^{-3} = \frac{5^{-21}}{3^{12}}\]

3. a⁰ = 1; if a ≠ 0

i.e. any non-zero number raised to the power zero is always equal to one (1).

e.g. 5⁰ = 1, 7⁰ = 1, (-8)⁰ = 1, (2^-5)⁰ = 1 and so on.

4. a^-m = \[\frac{1}{a^m}\] and \[\frac{1}{a^{-m}}\] = a^m; if a ≠ 0

e.g. 2^-3 = \[\frac{1}{2^3}\], 5^-7 = \[\frac{1}{5^7}\], \[\frac{1}{2^{-3}}\] = 2³, \[\frac{3^5}{2^3}\] and so on.

5. \[\sqrt[n]{a} = a^{\frac{1}{n}}\] and \[\sqrt[n]{a^m} = a^{\frac{m}{n}}\]

e.g. \[\sqrt{5} = 5^{\frac{1}{2}}\], \[\sqrt[6]{5^7} = 5^{\frac{7}{6}}\], \[\sqrt[3]{a^2 \times b^4} = a^{\frac{2}{3}} \times b^{\frac{4}{3}}\], etc.

Also remember that:

(i) (-a)^m = a^m; if m is even

e.g. (-5)⁴ = -5 × -5 × -5 × -5 = 5⁴

(ii) (-a)^m = -a^m; if m is odd.

e.g. (-5)³ = -5 × -5 × -5 = -5³

Test Yourself

8. (a²b^-3)⁴ = ..............................

9. (b^-4x²)^-2 = ......................................

10. (3x²y)² = ................................

11. \[\left(\frac{5m^2}{2n^3}\right)^3 = ......................................\]

12. \[\left(\frac{2a}{b^2}\right)^5 = ...............................\]

13. \[\left(x^{\frac{2}{3}} \cdot y^{\frac{-3}{2}}\right)^6 = ................................\]

14. (a²b)^-2.(ab)^-3 = .................................... = ..................

15. (125³)⁰ = .............

16. (-2)⁵ × (-2)³ = ................... = ..................

Teacher's Note

Exponent rules simplify calculations in physics and chemistry, such as measuring pH levels or calculating half-lives of radioactive materials.

Example 1:

Evaluate:

(i) \[4^{\frac{3}{2}} \times 125^{\frac{-2}{3}}\]

(ii) \[\left(\frac{8}{27}\right)^{\frac{2}{3}} + (32)^{\frac{-2}{5}}\]

(iii) -2⁴ - \[(\sqrt{3})^0\] × (-2)⁶ ÷ 4

Solution:

(i) \[4^{\frac{3}{2}} \times 125^{\frac{-2}{3}} = (2^2)^{\frac{3}{2}} \times (5^3)^{\frac{-2}{3}}\]

[4 = 2 × 2 = 2², 125 = 5 × 5 × 5 = 5³]

\[= 2^3 \times 5^{-2}\]

\[= \frac{8}{5^2}\]

\[= \frac{8}{25}\]

(Ans.)

\[2 \times \frac{3}{2} = 3 \text{ and } 3 \times \frac{-2}{3} = -2\]

\[2^3 = 2 \times 2 \times 2 = 8 \text{ and } 5^{-2} = \frac{1}{5^2}\]

(ii) \[\left(\frac{8}{27}\right)^{\frac{2}{3}} + (32)^{\frac{-2}{5}} = \left(\frac{2}{3}\right)^{3 \times \frac{2}{3}} + (2^5)^{\frac{-2}{5}}\]

\[\frac{8}{27} = \frac{2 \times 2 \times 2}{3 \times 3 \times 3} = \left(\frac{2}{3}\right)^3\]

and 32 = 2 × 2 × 2 × 2 × 2 = 2⁵

\[= \left(\frac{2}{3}\right)^2 + 2^{-2}\]

\[3 \times \frac{2}{3} = 2 \text{ and } 5 \times \frac{-2}{5} = -2\]

\[= \frac{2^2}{3^2} \times \frac{1}{2^{-2}}\]

\[\frac{1}{2^{-2}} = 2^2\]

\[= \frac{4}{9} \times 2^2\]

\[= \frac{4 \times 4}{9} = \frac{16}{9} = 1\frac{7}{9}\]

(Ans.)

(iii) Given expression

= -2⁴ - 1 × 2⁶ ÷ 2²

= -2⁴ - 2⁴

= -16 - 16 = -32

[(\(\sqrt{3}\))⁰ = 1; (-2)⁶ = 2⁶ and 4 = 2 × 2 = 2²]

[2⁶ ÷ 2² = 2^6-2 = 2⁴]

(Ans.)

Teacher's Note

Learning to evaluate complex exponential expressions builds problem-solving skills useful in fields like finance for calculating compound interest and in biology for modeling population growth.

Example 2:

Simplify: \[\frac{x^{m+n} \times x^{n+l} \times x^{l+m}}{(x^m \times x^n \times x^l)^2}\]

Solution:

Given expression = \[\frac{x^{m+n+n+l+l+m}}{x^{2m} \times x^{2n} \times x^{2l}}\]

= \[\frac{x^{2m+2n+2l}}{x^{2m+2n+2l}}\] = 1

(Ans.)

Example 3:

Simplify: \[\left(\frac{x^a}{x^b}\right)^{a+b} \times \left(\frac{x^b}{x^c}\right)^{b+c} \times \left(\frac{x^c}{x^a}\right)^{c+a}\]

Solution:

Given expression = (x^a-b)^{a + b} × (x^b-c)^{b + c} × (x^c-a)^{c + a}

= x^{(a-b)(a + b)} × x^{(b-c)(b + c)} × x^{(c-a)(c + a)}

= x^{a2 - b2} × x^{b2 - c2} × x^{c2 - a2}

= x^{a2 - b2 + b2 - c2 + c2 - a2}

= x⁰ = 1

(Ans.)

Teacher's Note

Simplifying complex exponent expressions like these is essential in algebra and calculus, where such manipulations appear frequently in real-world applications from engineering to computer science.

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