Uses of Exponents to Express Small Numbers in Standard Form

Small numbers are expressed using exponents to make them easy to calculate. Xn is the numeric expression where X is the base and n is the power or exponent. It is the representation of the number of times we need to multiply the base. The different laws of exponent are important so as to make the calculations efficient and effective. Exponents can be a fraction, whole number, decimals, or negative number. It is also known as the power of a number. 

Also read: Calculus Formula

Key Terms: Exponents, Small Numbers, Standard Form, Powers, Base, Large Numbers, Decimal

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Uses of Exponents to Express Small Numbers in Standard Form

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Suppose, the number is 0.000000006

To convert a small number into a standard form, we need to move the decimal point to the right. Place a multiplication sign after the single-digit and write the counted digits with a base 10 with a negative sign. 

0.000000006 = 61000000000 = 6109 = 6 x 10^-9 m

To express a large number of 150,000,000,000 in standard form, 

Write down the digits after the first digit followed by a decimal point and other numerical digits. Subtract the number of digits after the decimal point from the total number of digits. Place a multiplication sign after the digit and a negative sign as a power on the base of 10. 

150,000,000,000 = 1.5 x 10¹¹

Negative exponents are very effective to express small numbers in standard form. 

Also Read: Power and Exponents 

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Importance of Exponents in Expressing Numbers in Standard Form

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We convert large and small numbers in standard form, we use exponents as it makes it easier to read and compare. Any number written as a decimal number between 1.0 and 10.0n and multiplied by the power of 10, is known as the standard form of a number. 

Eg: 1.6 x 10^-3 and 0.19 x 10^-14 are standard forms

Standard Form of Exponents

Also Read: Integers as Exponents

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Large and Small Numbers

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Suppose the diameter of the sun is 1.4 x 10⁹ m and the diameter of the earth is 1.2756 x 10⁷ m.

Therefore, if compared 1.4 x 1091.2756 x 10⁷ = 1.4 x 109-71.2756 

= 1.4 x 1001.2756 

Thus, the diameter of the sun is about 100 times more than the diameter of the earth

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Laws of Exponents

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There are several laws of exponents to make calculations easier. 

Law 1: if the exponent of any number is 0 then the answer will be 1

X⁰ = 1

Law 2: if the exponent of any number is 1, then the answer will be the base itself

X¹ = X

Law 3: if the exponent of the number is -1, then the answer will be the reciprocal of the base

x^-1 = 1x

Law 4: if two numbers are there with the same base and different exponents, then the answer will be the addition of the exponents keeping the base the same

x^ax^b = x^a+b

Law 5: if two numbers with the same base and different exponents are to be divided, then the answer will be equal to the same base with the subtraction of the powers

x^a /x^b = x^a-b

Law 6: ( x^a )^b = x^ab

Law 7: if two numbers are to be multiplied with the same exponent, then each number will be calculated with the exponent 

(x*y)^a = x^a y^b

Law 8: if we have to divide two different bases with the base exponent, then the different bases will be divided individually with the same exponent

(x/y)^a = x^a/y^a

Law 9: if the exponent of any number is in the negative, then the answer will be the reciprocal of the base

x^-a = 1/x^a

Law 10: a¹/n = \(\sqrt{a}\)

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Things to Remember

• Very small numbers can be expressed in standard form by using negative exponents
• It is important to convert numbers with the same exponents when we need to add numbers in standard form
• If there are numerical expressions such as a, a^mx aⁿ = a^m+n, m and n are natural numbers
• Any number raised to the power of 0 is equal to 1. This is known as the zero power rule
• Multiplying the exponent values is known as the power of a power rule
• The number of times we have to multiply the base number is known as the exponents or the powers.
• If the exponent is in the fractional form, it is known as the fractional exponent rule
• Two different bases with the same power is known as a quotient to a power. 

Also read:

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