{"id":10309,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10309"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"exponent-rules-roots-and-equivalent-expressions","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/exponent-rules-roots-and-equivalent-expressions\/","title":{"rendered":"Exponent Rules, Roots And Equivalent Expressions"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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\n\n\n\n
\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Source: Kapdec.com<\/p>\n<\/div>\n

Unit: Exponents and Roots<\/strong><\/h2>\n

Exponent Rules, Roots and Equivalent Expressions<\/strong><\/h2>\n

First Law<\/strong><\/p>\n

If m<\/strong> is any non-zero rational number and p <\/strong>and q<\/strong> are natural numbers, then:<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 mp<\/sup> X mq<\/sup>= m p+q<\/sup><\/strong><\/p>\n

Generalization form of the above law:<\/p>\n

\u00a0<\/strong>If m<\/strong> is any non-zero rational number and p<\/strong>, q and r<\/strong> are natural numbers then:<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 mp <\/sup>X mq<\/sup> X mr<\/sup>= mp+q+r<\/sup><\/strong><\/p>\n

Example:\u00a0\u00a0 <\/strong>\u00a0Simplify and write the answer: 33<\/sup>\u00d732<\/sup>\u00d734<\/sup><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3\u00d73\u00d73\u00d73\u00d73\u00d73\u00d73\u00d73\u00d73 =39<\/sup>=33+2+4<\/sup><\/p>\n

\u00a0<\/p>\n

Dividing Powers with the Same Base:<\/strong><\/p>\n

Consider the following division:<\/p>\n

54<\/sup>\u00f752<\/sup>=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0= 5\u00d75=25<\/p>\n

Or 54<\/sup>\u00f752<\/sup>= 54-2<\/sup>=52<\/sup>= 25<\/p>\n

95 <\/sup>\u00f7 92<\/sup>=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0= 93<\/sup>=27<\/p>\n

Or 95 <\/sup>\u00f7 92<\/sup> = 95-2<\/sup>=93<\/sup>= 27<\/p>\n

m6<\/sup>\u00f7 m3<\/sup>=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0= m3<\/sup><\/p>\n

In all the above divisions of powers with the same base, we can say that the division of powers with the same base is equal to a power of the same base whose exponent is equal to the difference of exponent of numerator and denominator.<\/p>\n

Second Law:<\/strong><\/p>\n

If m<\/strong> is any non-zero rational number and p <\/strong>and q<\/strong> are natural numbers such that p>q, then<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 mp<\/sup>\u00f7mq<\/sup>= mp\u2500q <\/sup>or\u00a0<\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

= mp\u2500q<\/sup><\/strong><\/p>\n

Now students, from the above concept we can calculate following questions quickly:<\/p>\n

108<\/sup> \u00f7 103<\/sup> = 108 \u2013 3 <\/sup>= 105<\/sup><\/p>\n

79<\/sup> \u00f7 76<\/sup> = 79-6<\/sup> = 73<\/sup><\/p>\n

a8<\/sup> \u00f7 a5<\/sup> = a8\u25005<\/sup>= a3<\/sup><\/p>\n

\u00a0<\/p>\n

Example :\u00a0\u00a0 <\/strong>\u00a0Simplify and write the answer: 912<\/sup>\u00f7 99<\/sup><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 We have, 912<\/sup>\u00f7 99<\/sup> = 912\u25009<\/sup>= 93<\/sup>=243\u00a0\u00a0<\/p>\n

Taking Power of a Power:<\/strong><\/p>\n

We shall look at what is the power of a power. In order to understand the concept of the power of power Consider the following<\/p>\n

Simplify (33<\/sup>)2<\/sup> and (22<\/sup>)4<\/sup><\/p>\n

Now, (33<\/sup>)2<\/sup> means 33<\/sup> is multiplied two times with itself.<\/p>\n

(33<\/sup>)2<\/sup> = 33<\/sup> \u00d7 33<\/sup><\/p>\n

= 33 + 3 <\/sup>(Since am<\/sup> \u00d7 an<\/sup> = am <\/sup>+ <\/sup>n<\/sup>)<\/p>\n

= 36<\/sup> = 33 \u00d7 2<\/sup><\/p>\n

Thus (33<\/sup>)2<\/sup> = 33\u00d72<\/sup><\/p>\n

Similarly (22<\/sup>)4<\/sup><\/p>\n

(22<\/sup>)4<\/sup> = 22<\/sup> \u00d7 22<\/sup> \u00d7 22<\/sup> \u00d7 22<\/sup><\/p>\n

= 22 + 2 + 2 + 2<\/sup><\/p>\n

= 28<\/sup> (Observe 8 is the product of 2 and 4).<\/p>\n

= 22 \u00d7 4<\/sup><\/p>\n

Third Law:<\/strong><\/p>\n

From the above result we can generalize for any non-zero integer \u2018a\u2019, where \u2018m\u2019 and \u2018n\u2019 are whole numbers,<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (am<\/sup> )n<\/sup>= amn<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n

Example :\u00a0 <\/strong>Calculate the value of (24<\/sup>)5<\/sup><\/p>\n

\u00a0We have<\/p>\n

(24<\/sup>)5<\/sup> So from (am<\/sup>)n<\/sup>= amn<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n

\u00a0 (24<\/sup>)5<\/sup>= 24\u00d75<\/sup>= 220<\/sup><\/p>\n

Example: <\/strong>\u00a0Calculate (42<\/sup>)3<\/sup>\u00d7 (46<\/sup>)3<\/sup><\/p>\n

We have (42<\/sup>)3<\/sup>\u00d7 (46<\/sup>)3<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

So from\u00a0\u00a0 (am<\/sup>)n<\/sup>= amn<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

42\u00d73<\/sup>\u00d746\u00d73<\/sup><\/p>\n

46<\/sup>\u00d7418<\/sup><\/p>\n

46+18<\/sup>=424<\/sup><\/p>\n

Multiplying Powers with the Same Exponent:\u00a0\u00a0 <\/strong><\/p>\n

Students consider the following Products:<\/p>\n

23<\/sup> \u00d7 33<\/sup> = (2 \u00d7 2 \u00d7 2) \u00d7 (3 \u00d7 3 \u00d7 3)<\/p>\n

= (2 \u00d7 3) \u00d7 (2 \u00d7 3) \u00d7 (2 \u00d7 3)<\/p>\n

= 6 \u00d7 6 \u00d7 6<\/p>\n

= 63<\/sup> (Observe 6 is the product of bases 2 and 3)<\/p>\n

Consider 44<\/sup> \u00d7 34<\/sup> = (4 \u00d7 4 \u00d7 4 \u00d7 4) \u00d7 (3 \u00d7 3 \u00d7 3 \u00d7 3)<\/p>\n

= (4 \u00d7 3) \u00d7 (4 \u00d7 3) \u00d7 (4 \u00d7 3) \u00d7 (4 \u00d7 3)<\/p>\n

= 12 \u00d7 12 \u00d7 12 \u00d7 12<\/p>\n

= 124<\/sup><\/p>\n

Similarly, a4<\/sup> \u00d7 b4<\/sup> = (a \u00d7 a \u00d7 a \u00d7 a) \u00d7 (b \u00d7 b \u00d7 b \u00d7 b)<\/p>\n

= (a \u00d7 b) \u00d7 (a \u00d7 b) \u00d7 (a \u00d7 b) \u00d7 (a \u00d7 b)<\/p>\n

= (a \u00d7 b)4<\/sup><\/p>\n

= (ab)4<\/sup><\/p>\n

\u00a0It is clear from the above that the product of powers with different bases and same exponents is equal to the power whose base is equal to the product of different bases and exponent equal to the common exponent.<\/p>\n

Fourth Law:<\/strong><\/p>\n

If p, q<\/strong> are non-zero rational numbers and \u201cn\u201d <\/strong>is a natural number, then <\/strong><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 pn<\/sup>\u00d7qn<\/sup>= (pq)n<\/sup><\/strong><\/p>\n

Generalization: If p, q<\/strong> and r <\/strong>are non zero rational numbers and \u201cn\u201d <\/strong>is a natural number, then <\/strong><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0pn <\/sup>\u00d7 qn <\/sup>\u00d7 rn<\/sup>= (pqr)n<\/sup><\/strong><\/p>\n

Example: <\/strong>Express the following products of powers as the exponent of a rational number: 43<\/sup>\u00d763<\/sup><\/p>\n

We have,<\/p>\n

43<\/sup>\u00d763<\/sup><\/p>\n

So, from pn<\/sup>\u00d7qn<\/sup>= (pq)n<\/sup><\/strong><\/p>\n

43<\/sup>\u00d763<\/sup>= (24)3<\/sup>=13824<\/p>\n

Dividing Powers with the Same Exponent<\/strong>:\u00a0\u00a0 <\/strong><\/p>\n

Consider the following Simplifications:<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0=<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0<\/p>\n

Consider\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=\u00a0\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0\u00a0 = <\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

Consider, also,\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0=\u00a0\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0= <\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

Similarly,\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0=<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0It is clear from the above that the division of powers with the same exponents is equal to the power whose base is equal to the division of bases and exponent equal to the same exponent.<\/p>\n

Fifth Law:<\/strong><\/p>\n

If p, q<\/strong> are non-zero rational numbers and \u201cn\u201d <\/strong>is a natural number, then <\/strong><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=\u00a0<\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

Example: <\/strong>Simplify\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

We have,\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

So, from, <\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=\u00a0<\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=\u00a0<\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=<\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0=\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0<\/p>\n

Algebra: Exponents and Powers<\/strong><\/p>\n

Laws of Exponents ON Negative Powers:<\/u><\/strong><\/p>\n

\u00a0<\/p>\n

We have learned the following laws of exponents of rational numbers when exponents are whole numbers.<\/p>\n

\u00a0<\/p>\n

(i)mp<\/sup><\/strong> <\/sup><\/strong>\u00d7 mq<\/sup>= m p+q<\/sup><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong>(First Law)<\/p>\n

(ii) <\/strong>mp<\/sup>\u00f7mq<\/sup>= mp\u2500q<\/sup><\/strong>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/sup><\/strong><\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

=mp\u2500q<\/sup><\/strong> , p>q<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(Second Law)<\/p>\n

(iii) <\/strong>(am<\/sup><\/strong> )<\/strong>n<\/sup><\/strong>= amn<\/sup><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>(Third Law)<\/p>\n

\u00a0(iv) pn<\/sup>\u00d7qn<\/sup>= (pq)n<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>(Fourth Law)<\/p>\n

(v)\u00a0<\/sup><\/strong>\u00a0\u00a0\u00a0<\/p>\n

\"\"<\/p>\n

Source: Kapdec.com<\/p>\n<\/div>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (Fifth Law)<\/p>\n

These laws also hold good for negative integral exponents<\/strong>.<\/p>\n

Techniques for Creating Equivalent Expressions:<\/strong><\/p>\n

    \n
  • Rearranging Like Terms:<\/strong> Just like rearranging ingredients in a recipe doesn’t change the final product, you can rearrange terms (numbers and variables) within an expression as long as you don’t change their order of operation (+, -, x, \/).\n
      \n
    • Example: 2x + 3y is equivalent to 3y + 2x<\/li>\n<\/ul>\n<\/li>\n
    • The Commutative Property:<\/strong> This property applies to addition and multiplication, stating that the order doesn’t affect the outcome.\n
        \n
      • Addition: a + b = b + a (e.g., 5 + 7 = 7 + 5)<\/li>\n
      • Multiplication: a x b = b x a (e.g., 3 x 4 = 4 x 3)<\/li>\n<\/ul>\n<\/li>\n
      • The Associative Property:<\/strong> This property applies to addition and multiplication, allowing you to group terms differently without affecting the result.\n
          \n
        • Addition: (a + b) + c = a + (b + c) (e.g., (2 + 3) + 4 = 2 + (3 + 4))<\/li>\n
        • Multiplication: (a x b) x c = a x (b x c) (e.g., (2 x 3) x 5 = 2 x (3 x 5))<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

          Tips and Tricks:<\/strong><\/p>\n

            \n
          • Factoring and Expanding:<\/strong> Sometimes, an expression can be simplified by factoring out common terms or expanding parentheses using the distributive property (a(b + c) = ab + ac). Recognizing equivalent forms after factoring or expanding is crucial.<\/li>\n
          • Practice Makes Perfect:<\/strong> The more you practice manipulating expressions, the better you’ll become at recognizing equivalent forms.<\/li>\n
          • \n
            \"\"<\/p>\n

            Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n<\/ul>\n

            <\/p>\n