Question

Laws of integral exponents

Answer 1

Section 5-1: Integral Exponents BasicLaws of Exponents Each of the above rules should be familiar to you from algebra I. Here are some sample problems with their solutions. The first one is squaring a negative number and the second is squaring a positive number and then making the whole result negative.

Answer 2

Question:-

• Laws of exponents (for integral purpose)

Answer:-

1. Product law :- $\sf{a^{m} \times a^{n}=a^{m+n}}$

• For example:-

   $\sf⇒{3^{3} \times 3^{5}=3^{3+5}=3^{8};5^{8} \times 5^{5}=5^{8+5}=5^{13}}$

   $\sf{⇒ 7^{2} \times 7^{4}=7^{2+4}=7^{6};2^{-5} \times 2^{8}=2^{-5+8}=2^{3} \: and \: so \: on$

2. Quotient law :-

  $\sf{\dfrac{a^{m}}{a^{n}}=a^{m-n}, \: if \: m>n \: \: \: \: \: \: and \: \: \: \: \dfrac{a^{m}}{a^{n}}=\dfrac{1}{a^{n-m}}, \: if \: n>m$

• For example:-

$\sf{⇒\dfrac{2^{12}}{2^{7}}=2^{12-7}=2^{5};\dfrac{2^{6}}{2^{13}}=\dfrac{1}{2^{13-6}}=\dfrac{1}{2^{7}};\sf{\dfrac{5^{12}}{5^{-3}}=5^{12+3}=5^{15};\dfrac{3^{-6}}{3^{3}}=\dfrac{1}{3^{3+6}}=\dfrac{1}{3^{9}} \: and \: so \: on.}$

3. Power law :- $\sf{(a^{m})^n=a^{mn}$

$\sf{⇒(3^{5})^2=3^{5 \times 2}=3^{10};(5^{6})^-3=5^{6 \times -3}=5^{-18} ;\sf{(7^{-2})^3=7^{-2 \times 3}=7^{-6};(5^{-3})^-2=5^{6} \: and \: so \: on$

Know more:-

$\sf{(-2)}^{3}=-2 \times -2 \times -2 = -8,}\\\sf{(-2)}^{4}=-2 \times -2 \times -2 \times -2=16,}\\\sf{(-2)}^{5}=-2 \times -2 \times -2 \times -2 \times -2=-32}\\\sf{(-2)^{6}=-2 \times -2 \times -2 \times -2 \times -2 \times -2=64 \: \: and \: so \: on$

Thus,

$\sf{(i) \: If \: n \: is \: even, \: (-2)^{n} \: is \: positive.}\\\sf{(ii)If \: m \: is \: odd, \: (-2)^{n} \: is \: negative}\\\sf{\: \: \: \: \: In \: general \: (-a)^{n}=a^{n}, if \: n\: is \: even}\\\sf{and, \: \: \: \: (-a)^{n}=-a^{n}, if \: n \: is \: odd}$