Key Laws of Exponents and Powers with Practical Applications
Exponents and powers are a combined simplification form of multiplication. By using exponent and power, we can solve multiplication easily. Power is an expression, which represents the repeated multiplication of a number. The factor, which is multiplied repeatedly, is called the base. The number of multiplications is called the exponent. For example, 5 is multiplied 3 times. So, 5* 5* 5 = 53, where 5 is base, 3 is exponent, and 53 is power. This expression 53 is known as 5 to the power of 3. Thus, you can take any multiplication factor to multiply the same factor multiple times and convert into power expression. After that, you will be able to identify the base and exponent from the expression.
General Form of Power and Exponent
Power and exponent are used to simplify the multiplication of the same factor. The power expression represents the whole process and the exponent defines the time of the factor to be multiplied. If the factor a, is multiplied for n times, the power expression is an. Here, an is expressed as:
an = a* a* a* ……. * a (n times)
Hence, we can say that it is a simplified method of repeated multiplication.
Laws of Power and Exponent
The laws of exponents and powers are mainly based on the exponent part. Therefore, the laws are also known as the laws of exponents. The laws are defined below with expression.
Multiplication Law
This law defined that, when two exponents are multiplied with the same base factor, the two exponents are added in the result. The base or multiplication factor remains the same. The expression of multiplication law is:
am * an = am+n.
Division Law
As per division law, when two exponents are divided with the same base factor, the two exponents are subtracted in the result. The base will remain the same. The expression of this law is:
am / an = am-n.
Negative Exponent Law
As per negative exponent law, if any base has a negative exponent, it comes in the reciprocal with positive power to the base. The expression of negative exponent law is given below as:
a-n = 1/an
Rules of Exponents and Powers
There are several specific rules of exponents and powers. The rules are dependent on the laws of exponents. Here, we have given a brief explanation of the rules. Suppose, x and y are two integers and p and q are the exponents. Now, the rules are:
If the power of any integer factor is o, the result will be one. The expression of this rule is x0 = 1. For example, 20 = 1.
If the exponent of a power expression is an integer, the result exponent is the product of two exponents. The expression is, (xp)q = xpq. For example, (122)3 = 122*3 = 126.
If the two different bases are multiplied with the same exponent, the result is the product of two bases and the exponent remains the same. The expression of this rule is xp * yp = (xy)p. For example, 22*32 = (2*3)2 = 62.
If two different bases are divided with the same exponent, the result is the division of two bases and the exponent remains the same. The expression is, xp / yp = (x/y)p. For example, 22 / 32 = (2/3)2.
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Applications of Exponents and Powers
In the scientific field, these are the two most applied concepts. There are huge numerical calculations in different scientific fields. That is why exponents are necessary for scientific calculations. The necessity of powers and components can be explained with examples. The mass of the sun is measured as 1,989,000,000,000,000,000,000,000,000,000 Kg. The distance between the Earth and the Sun is measured to be 149,600,000 Km. These are huge numbers to memorize as well as to calculate. With the help of an exponent, these numbers can be broken into powers of 10. After using exponent, the distance between earth and Sun becomes 1.496 * 108 Km and the mass of the Sun becomes 1.989 * 1030 Kg.
Solved Examples
1. Identify the base, exponent, and power of the following expressions -
(16)2 b) (31)3
Solution:
The base is 16.
The exponent is 2.
The power is (16)2.
The base is 31.
The exponent is 3.
The power is (31)3.
2. Convert the multiplications into powers:
5 × 5 × 5 × 5 × 5 × 5
3 × 3 × 3 × 3 × 3 × 3 × 3 × 3
Solution:
5 × 5 × 5 × 5 × 5 × 5 = 56.
3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 38.
FAQs on Exponents and Powers Explained for Students
1. What are exponents and powers, and how are they represented?
Exponents and powers provide a shorthand way to write repeated multiplication. In an expression like aⁿ, 'a' is the base (the number being multiplied), 'n' is the exponent (how many times the base is multiplied by itself), and the entire expression aⁿ is called the power. For example, instead of writing 5 x 5 x 5, we can simply write 5³, where 5 is the base, 3 is the exponent, and 5³ is the power.
2. What are the main laws of exponents used for simplification?
The main laws of exponents are fundamental rules for simplifying expressions. As per the CBSE syllabus, the key laws are:
- Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a Power Rule: (aᵐ)ⁿ = aᵐⁿ
- Product of Powers Rule: aᵐ × bᵐ = (ab)ᵐ
- Quotient of Powers Rule: aᵐ ÷ bᵐ = (a/b)ᵐ
- Zero Exponent Rule: a⁰ = 1 (for any non-zero 'a')
- Negative Exponent Rule: a⁻ⁿ = 1/aⁿ
3. What does a negative exponent signify in mathematics?
A negative exponent indicates a reciprocal. Instead of multiplication, it signifies repeated division. According to the rule a⁻ⁿ = 1/aⁿ, a number raised to a negative exponent is equal to 1 divided by that number raised to the positive exponent. For instance, 4⁻² is not a negative number; it is the reciprocal of 4², which means 1 / (4 x 4) or 1/16.
4. How is any non-zero number with an exponent of zero equal to 1?
This can be understood using the quotient rule of exponents. Consider the expression aᵐ ÷ aᵐ. Since any number divided by itself is 1, the result is 1. Now, applying the quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ) to the same expression, we get aᵐ⁻ᵐ = a⁰. Since both expressions are equal, it proves that a⁰ = 1 for any non-zero number 'a'. This rule is a logical consequence of the other exponent laws.
5. Why is understanding exponents and powers important in real life?
Understanding exponents is crucial because they are used to model and calculate phenomena involving very large or very small quantities. Real-world applications include:
- Science: Expressing astronomical distances (e.g., light-years) or the size of microscopic organisms in standard form.
- Finance: Calculating compound interest over multiple periods.
- Technology: Measuring computer memory (kilobytes, megabytes, gigabytes are all powers of 2).
- Population Growth: Modelling exponential growth of populations.
6. How do you express very large or small numbers in standard form using exponents?
To express numbers in standard form (or scientific notation), we write them as a product of a number between 1 and 10, and a power of 10. The format is k × 10ⁿ, where 1 ≤ k < 10. For a large number like 300,000,000 m/s (speed of light), we move the decimal 8 places to the left to get 3.0, so the standard form is 3.0 × 10⁸ m/s. For a small number like 0.0005, we move the decimal 4 places to the right to get 5.0, so the standard form is 5.0 × 10⁻⁴.
7. What is the common mistake when comparing expressions like (2³)⁴ and 2³⁴?
The common mistake is to assume they are the same. They represent vastly different calculations.
- (2³)⁴ follows the 'power of a power' rule, meaning you multiply the exponents: 2⁽³*⁴⁾ = 2¹², which equals 4096.
- 2³⁴ means 2 raised to the power of (3⁴). First, you calculate the exponent 3⁴ (which is 3×3×3×3 = 81). The expression then becomes 2⁸¹, an immensely larger number.
8. Can the base of an exponent be a fraction or a decimal?
Yes, absolutely. The laws of exponents apply to any real number as the base, not just integers. For example, if the base is a fraction like (1/2)³, it means (1/2) × (1/2) × (1/2), which equals 1/8. Similarly, with a decimal base like (0.5)², it means 0.5 × 0.5, which equals 0.25. This concept is important for calculations in various scientific and financial fields.
