Class 8 — Mathematics (NCERT)

Exponents and Powers

Class 8

✓By the end of this lesson students will be able to understand the meaning of negative integral exponents.
✓By the end of this lesson students will be able to apply the laws of exponents to simplify expressions involving negative exponents.
✓By the end of this lesson students will be able to express very small and very large numbers in standard form.
✓By the end of this lesson students will be able to convert numbers from standard form to their usual form.

Key concepts

Exponents (A Quick Recap)

An exponent tells us how many times a base number is multiplied by itself. For example, in a^n, 'a' is the base and 'n' is the exponent. a^n = a x a x a x ... (n times). We have previously studied exponents with positive integral powers.

Negative Integral Exponents

So far, we have dealt with positive integral exponents. What if the exponent is a negative integer? For any non-zero rational number 'a' and a positive integer 'm', a^(-m) is the multiplicative inverse of a^m. This means that a^(-m) is equal to 1 divided by a^m.

a^(-m) = 1/a^m (where a ≠ 0)

Laws of Exponents (Extended to Negative Exponents)

The laws of exponents that we learned for positive integral exponents also hold true for negative integral exponents. For any non-zero rational numbers 'a' and 'b', and any integers 'm' and 'n', the following laws apply:

a^m × a^n = a^(m+n)\na^m / a^n = a^(m-n)\n(a^m)^n = a^(mn)\n(ab)^m = a^m × b^m\n(a/b)^m = a^m / b^m\na^0 = 1 (where a ≠ 0)

Standard Form (Scientific Notation)

When we deal with very large or very small numbers, it is convenient to express them in a special form called standard form. A number is said to be in standard form if it is expressed as k × 10^n, where k is a decimal number such that 1 ≤ k < 10, and n is an integer.\n\nFor very large numbers, n will be a positive integer.\nFor very small numbers (between 0 and 1), n will be a negative integer.

k × 10^n (where 1 ≤ k < 10 and n is an integer)

Key facts to remember

1For any non-zero rational number 'a' and positive integer 'm', a^(-m) = 1/a^m.
2The laws of exponents (a^m × a^n = a^(m+n), a^m / a^n = a^(m-n), etc.) are applicable for all integers m and n.
3Any non-zero number raised to the power of zero is 1 (i.e., a^0 = 1, where a ≠ 0).
4A number is in standard form if it is written as k × 10^n, where 1 ≤ k < 10 and n is an integer.
5For very small numbers (between 0 and 1), the exponent 'n' in standard form is negative.
6For very large numbers (greater than 10), the exponent 'n' in standard form is positive.

Worked examples

Example 1

Simplify: (3^(-2) × 4^(-2)) / 2^(-3)

IGiven expression: (3^(-2) × 4^(-2)) / 2^(-3)

IIUsing the law a^(-m) = 1/a^m, we rewrite the terms:

III= (1/3^2 × 1/4^2) / (1/2^3)

IVCalculate the powers:

V= (1/9 × 1/16) / (1/8)

VIMultiply the fractions in the numerator:

VII= (1/(9 × 16)) / (1/8)

VIII= (1/144) / (1/8)

9To divide by a fraction, multiply by its reciprocal:

10= 1/144 × 8/1

11= 8/144

12Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (8):

13= 1/18

Answer

1/18

Always convert negative exponents to positive exponents by taking the reciprocal of the base raised to the positive power. This often simplifies calculations.

Example 2

Express 0.0000000085 in standard form.

IThe given number is 0.0000000085.

IITo express a number in standard form (k × 10^n), we need to move the decimal point such that there is only one non-zero digit to its left. In this case, we move the decimal point to the right.

IIIMove the decimal point 9 places to the right to get 8.5.

IVSince the decimal point was moved 9 places to the right, the exponent of 10 will be -9 (for numbers between 0 and 1, the exponent is negative).

VSo, 0.0000000085 = 8.5 × 10^(-9).

Answer

8.5 × 10^(-9)

When expressing a very small number (between 0 and 1) in standard form, the exponent 'n' will always be a negative integer.

Example 3

Express 1,27,50,000 in standard form.

IThe given number is 1,27,50,000. This can be written as 12750000.0 (with an implied decimal point at the end).

IITo express this number in standard form (k × 10^n), we need to move the decimal point such that there is only one non-zero digit to its left. In this case, we move the decimal point to the left.

IIIMove the decimal point 7 places to the left to get 1.275.

IVSince the decimal point was moved 7 places to the left, the exponent of 10 will be +7 (for numbers greater than 10, the exponent is positive).

VSo, 1,27,50,000 = 1.275 × 10^7.

Answer

1.275 × 10^7

When expressing a very large number (greater than 10) in standard form, the exponent 'n' will always be a positive integer.

Common mistakes

✗Confusing a^(-m) with -a^m. Remember, a^(-m) is 1/a^m, not the negative of a^m.
✗Incorrectly applying the laws of exponents, especially with signs. For example, (a^m)^n is a^(mn), not a^(m+n).
✗Making errors in counting the number of decimal places when converting to or from standard form, leading to an incorrect exponent 'n'.
✗Forgetting that a^0 = 1 is only valid when the base 'a' is non-zero. 0^0 is undefined.
✗Not ensuring that the 'k' part of the standard form (k × 10^n) is strictly between 1 and 10 (i.e., 1 ≤ k < 10).

Exam tips

★Always convert negative exponents to positive exponents first using a^(-m) = 1/a^m to simplify calculations and avoid sign errors.
★Memorise all the laws of exponents thoroughly and practice applying them to various problems. Understanding the laws is crucial for solving complex expressions.
★When converting to standard form, clearly count the number of places the decimal point is shifted. Remember, shifting the decimal point to the right results in a negative exponent, and shifting to the left results in a positive exponent.
★Show all steps clearly in your solutions, especially when simplifying expressions involving multiple laws of exponents. This helps in identifying and correcting any errors and ensures you get partial marks even if the final answer is incorrect.

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