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Exponents and Powers

    Laws of Exponents

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8th class > Exponents and Powers > Laws of Exponents

Laws of Exponents

We have learnt that for any non-zero integer a , am×an=am+n, where m and n are natural numbers.

Example 2

Does this law also hold if the exponents are negative?

(i) Let us explore. 2−3 and 2−2

2−3and2−2

  • We know that 2−3 = and 2−2 =
  • Numerators and denominators are same, so add the powers
  • We get the result as
  • If we bring denominator to numerator then the power will be

(ii) Take −3−4×−3−3

−3−4×−3−3

  • We know that −3−4 = and =
  • Numerators and denominators are same so, add the powers
  • We get the result as
  • If we bring denominator to numerator then the power will be

(iii) Now consider 5−2×54

5−2×54

  • we know that 5−2 =
  • Divide the exponents
  • Numerators and denominators are same so, subtract the powers
  • We get the result
  • If we bring denominator to numerator then the power will be

In general, we can say that for any non-zero integer am×an=am+n, where m and n are integers.

Try these.

Match the below exponents

(–2)^(–3) × (–2)^(–4)
p^3 × p^(–10)
3^2 × 3^(-5) × 3^(6)
3^3
-2^-7
p^-7

On the same lines you can verify the following laws of exponents, where a and b are non zero integers and m, n are any integers.

(i) aman=am−n(ii) amn=amn(iii) am×bm=abm
(iv) ambm=abm(v) a0=1

Examples on Laws OF Exponents

Let us solve some examples using the above Laws of Exponents.

Example 3: Find the value of

(i) 2−3 =
2−3 = 123 =
(ii) 13−2=
13−2 = = 3 × 3 =

Example 4: Simplify

(i) −45×−4−10 =
Note : (am×an=am+n)
−45−10 = = 1/(-4)^
(ii) 25÷2−6=
Note : (am÷an=am−n)
25−−6 =

Note :an= 1 only if n = 0.

This will work for any a. For a = 1, 11= 12= 13 = ... = 1 or 1n=1 for infinitely many n.

For a = –1, −10= −12= −14= −1−2 = ... 1 or −1p = 1 for any even integer p.

Example 5:

Simplify and write the answer in the exponential form.

simply the exponential form

  • Here exponential form is 25÷285×2−5
  • 2 is the Base of all terms so, subtract the powers in the bracket. We get the result is
  • Bases are same so, multiply the powers
  • we get the result is
  • converting the exponent:

Example 6:

Find m so that −3m+1×−35=−37

Find value of m

  • Here is the exponent is −3m+1×−35=−37
  • Bases are same then add the exponents,we get power is
  • On both the sides, we have the same base, so their exponents must be equal.
  • subtracting the values
  • Hence m is

Example 7:

Find the value of 23−2

Solution:

23−2=2−23−2=3222=

In general, ab−m=bam