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Mathematics

State True or False :

(i) For any non-zero number a, we have (am)n = am+n.

(ii) If a and b are non-zero numbers, then {(ab)m}n=(ba)−mn\Big\lbrace\Big(\dfrac{a}{b}\Big)^m\Big\rbrace^n = \Big(\dfrac{b}{a}\Big)^{-mn}

(iii) For a non-zero number x; (xm x xn) is equal to x to the power (m + n).

(iv) If a number p is multiplied n times, then the resulting number is pn.

(v) In an exponential notation xn; n is called the index.

Exponents

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Answer

(i) False
Reason — The power of a power law states that (am)n = am x n. The expression am+n is the result of multiplying powers with the same base (am x an).

(ii) True
Reason — According to the power of a power rule, {(ab)m}n=(ab)mn\Big\lbrace\left(\dfrac{a}{b}\right)^m\Big\rbrace^n = \left(\dfrac{a}{b}\right)^{mn}. By applying the reciprocal rule (ab)mn=(ba)−mn\left(\dfrac{a}{b}\right)^{mn} = \left(\dfrac{b}{a}\right)^{-mn}, the statement is mathematically correct.

(iii) True
Reason — This is the product law of exponents, which states that for any non-zero base x, xm x xn = x(m+n).

(iv) True
Reason — By definition, exponential notation is a shorthand for repeated multiplication; if p is multiplied n times, it is written as pn.

(v) True
Reason — In the notation xn, the number n is commonly referred to as the exponent, power, or index.

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