Indices

Laws of Indices

Laws of Indices (Exponents)

Indices (or exponents) are used to express repeated multiplication of a number. If a is a real number and m,n are integers, then the laws of indices are as follows:

1. Product Law

$a^m\times a^n=a^{m+n}$

a^m × a^n = a^(m + n)

Example:

$2^3\times 2^4=2^{3+4}=2^7$

2^3 ×2^4 = 2^(3 + 4)

=2^7

2. Quotient Law

$\frac{a^m}{a^n}=a^{m-n},\text{where }a\mathrm{\ne }0$

a^n / a^m​ = a^(m − n), 

where a = 0

Example:

$\frac{5^6}{5^2}=5^{6-2}=5^4$

5^2/ 5^6​ = 5^(6 − 2) = 5^4

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3. Power of a Power Law

$(a^m{)}^n=a^{m\times n}$

(a^m)^n=a^(m×n)

Example:

$(3^2{)}^4=3^{2\times 4}=3^8$

(3^2)^4 = 3^(2×4) = 3^8

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4. Power of a Product Law

$(ab{)}^m=a^m\times b^m$

(ab)^m = a^m × b^m

Example:

$(2\times 3{)}^4=2^4\times 3^4$

(2×3)^4 = 2^4 × 3^4

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5. Power of a Quotient Law

${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m},\text{where }b\mathrm{\ne }0$

(b/a​)^m = b^m/a^m​,

where b=0

Example:

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

(5/4​)^3 = 5^3/4^3 ​= 125 / 64​

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6. Zero Index Law

$a^0=1,\text{where }a\mathrm{\ne }0$

a^0 = 1,

where a=0

Example:

$7^0=1$

7^0 = 1

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7. Negative Index Law

$a^{-m}=\frac{1}{a^m},\text{where }a\mathrm{\ne }0$

a^−m = 1/a^m​,

where a=0

Example:

$3^{-2}=\frac{1}{3^2}=\frac{1}{9}$

3^−2 = 1/3^2 ​= 1/9​

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8. Fractional Indices (Roots Law)

$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a}{)}^m$

a^(m/n) ​= (n√a)^m

​

Example:

${16}^{\frac{1}{2}}=\sqrt{16}=4$

16^(1/2) ​= √16.  ​=4

${27}^{\frac{2}{3}}=(\sqrt[3]{27}{)}^2=3^2=9$

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Examples Using Multiple Laws

1. Simplify 

   $2^3\times 2^{-1}$

   • Using the product law:

     $2^{3+(-1)}=2^2=4$

2. Simplify 

   $\frac{5^4}{5^6}$

   • Using the quotient law:

     $5^{4-6}=5^{-2}=\frac{1}{5^2}=\frac{1}{25}$

3. Simplify 

   $(4^2{)}^3$

   • Using the power of a power law:

     $4^{2\times 3}=4^6=4096$