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The laws of exponents are rules that represent expressions in simpler terms. They are used to solve problems that require repeated multiplication processes.
- Laws of exponents are used to simplify multiplication and division operations, aiding in problem-solving.
- These laws are also popular with the term exponent rules.
- The laws of exponents can easily solve problems involving the use of complex powers such as fractions, decimals, and roots.
- Exponents, also known as powers, refers to the number of times an individual digit or number is multiplied.
- For illustration, 5 × 5 × 5 can be written in the form 5³.
- In this case, the exponent is 3, which indicates the number of times 5 has to be multiplied.
| Table of Content |
Key Terms: Laws of Exponents, Exponent Rules, Product Law of Exponents, Quotient Law of Exponents, Zero Law of Exponents, Identity Exponent Law, Negative Law of Exponents, Power of a Power Law of Exponents, Power of Product Rule of Exponents, Rule of Exponents
What are Exponents?
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Exponents indicate how many times we have to multiply the base number in terms of the power of the number. They are also known as powers or indices.
- Suppose you are given the number 10,000,000,000,000, which is a huge, massive number that is not easy to read, recognize, and evaluate.
- With the help of exponents, we can express numbers in a form that is easy to read, recognize and evaluate.
- Suppose a number ‘y’ is multiplied by z a number of times. Then it is written as yz, with x as the base and z as the exponent.
- It can be represented as yz = y × y × y × y × y × y × y ……….. × y (z times).
- In case the exponent of a number is 2 then it is called squared.
- However, in the case of the exponent of number 3, it is called cubed.
- For illustration, we can write the mass of electrons as 9.1×10-31 kg.
Exponents ExamplesExample: The exponents examples are as follows:
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Also Read:
| Related Articles | ||
|---|---|---|
| Power and Exponents | Exponential Formula | Exponent Rules |
| Types of Numbers | Standard Form Formula | Exponent and Power MCQ |
Laws of Exponents
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The laws of exponents state rules and procedures for solving arithmetic problems easily. It is also known as the properties of exponents.
- Laws of exponents help simplify complex expressions.
- Some of the important exponent's laws are as follows:
Product Law of Exponents
The product law of exponents is applied to duplicate expressions sharing the same base value. According to this rule, "When multiplying with the same base, combine the exponents while keeping the base constant.
\(a^m × a^n = a^{m+n}\)
Product Law of Exponents ExampleExample: 23 × 26 = 23+6 =29 |
Quotient Law of Exponents
The quotient property of exponents is utilized to untangle expressions with a common base. This property stipulates, "When dividing two expressions sharing the same base, subtract their exponents while keeping the base constant.
\(\frac{a^m}{a^n} = a^{m-n}\)
Quotient Law of Exponents ExampleExample: 24/22 = 24-2 = 22 |
Zero Law of Exponents
The zero law of exponents is relevant exclusively when the exponent is applied to the base value 0. According to this principle, "Any number (excluding 0) raised to the power of 0 equals 1." It's important to note that 0^0 is undefined.
\(a^0 = 1\)
Zero Law of Exponents ExampleExample: 50 = 1 |
Identity Exponent Law
Identity exponent law indicates raising any number to an exponent of one is equivalent to the number itself.
\(a^1 = a\)
Identity Exponent Law ExampleExample: 21 = 2 |
Negative Law of Exponents
The Negative Law of the Exponents rule applies specifically when the provided exponent is a negative number. The principle declares that to transform any negative exponent into a positive exponent, it should be reciprocated by dividing it into one.
\(a^{-m} = \frac{1}{a^m}\; or \;(\frac{a}{b})^{-m} = (\frac{b}{a})^m\)
Negative Law of Exponents ExampleExample: 3-6 = 1/36 |
Power of a Power Law of Exponents
Power of a Power Law of Exponents rule is relevant to expressions in the form of (am)n. According to this rule, "When dealing with a single base and two exponents, simply multiply the powers.
\((a^m)^n = a^{mn}\)
Power of a Power Law of Exponents ExampleExample: (24)2 = 24*2 = 28 |
Power of Product Rule of Exponents
The rule of exponents for the power of a product is akin to the distributive property in mathematics. In this case, two or more bases have the same power.
\((ab)^m = a^m \times b^m\)
Power of Product Rule of Exponents ExampleExample: (5×5)4 = 54 x 54 |
Quotient to a Power
The power of the quotient law operates similarly to the power of product exponents, with the distinction that the base value is presented in the form of a division.
\((\frac{a}{b})^m = \frac{a^m}{b^m}\)
Quotient to a Power ExampleExample: (2/6)2 = 22/62 |
Exponent Rules Chart
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The exponent rules chart for various laws of exponents are as follows:
| Name of Exponent Rules | Rule |
|---|---|
| Zero Exponent Rule | a0 = 1 |
| Identity Exponent Rule | a1 = a |
| Product Rule | am× an = am+n |
| Quotient Rule | am/an = am-n |
| Negative Exponents Rule | a-m = 1/am |
| Power of a Power Rule | (am)n = amn |
| Power of a Product Rule | (ab)m = ambm |
| Power of a Quotient Rule | (a/b)m = am/bm |
Things to Remember
- Laws of exponents or properties of exponents are used to simplify involving exponents more easily.
- If you raise a negative number to even power, then the result of the expression is positive.
- However, in the case of odd numbers, the power result is always negative.
- If any number except one is raised to the power of infinity, then the result is always infinity.
- It is used by many experts like bankers, biologists, financial advisors, insurance risk engineers, assessors, and computer programmers.
- Some real-life applications of laws of exponents include radioactive decay, population growth, and loan interest rates.
Also Read:
Sample Questions
Ques: Solve the exponent (42)5? (2 marks)
Ans: By using power of exponent rule; (am)n = amn
⇒ (42)5 = 42×5
∴ the result is 410
Ques: Solve the exponent 64/6? (2 marks)
Ans: The quotient property of the objects is used to separate the expressions involving the same base. The key condition is that both the dividend and divisor must share the same bases.
⇒ Using division rule of exponents; am/an = am-n
⇒ 64/6
= 64-1
=63
∴ the result is 216
Ques: Simplify the exponent 3-2? (2 marks)
Ans: By applying the negative exponent rule; a-m = 1/am or (a/b)-m = (b/a)m
=3-2
∴ the result is 1/9
Ques: Simplify using laws of exponents (3√8)-2? (3 marks)
Ans: We can convert an expression with roots into an exponent using the following property: a1/n = n√a (3√8)-2 can be written as
=8 -2/3 ]
=1/8 2/3 [Applying negative law of exponents]
=1/641/3
∴ the result is 1/3√64
Ques: Solve the exponents 3-6 × (−5)-6? (2 marks)
Ans: By applying the power of same exponent law; (ab)m = am x bm
= 3-6 ×(−5)-6 = (3×(−5))-6
∴ the result is (-15)-6
Ques: Solve the exponents 4-3 × (−2)-3 + 5-3? (3 marks)
Ans: Considering the BODMAS rule
=(4-3 × (−2)-3 )+ 5-3
=(1 × (-2))-3 + 5-3
=-2-3 + 2-3
=-2-1-3
= -3-3
∴ the result is -6
Ques: Find the value of 10-4/10-2? (2 marks)
Ans: Given 10-4/10-2
= 10-4-(-2)
= 10-4+2
= 10-2
= 1/102
∴ the result is 1/100
Ques: Express 164 as a power raised to base 2? (2 marks)
Ans: We have 2×2×2×2 = 24
Therefore (24)4 = 216
Ques: Simplify and Write the Exponential Form of 1/16×5-4? (2 marks)
Ans: We can write 1/16 as 2-4
= 2-4×5-4
= (2×5)-4
∴ the result is (10)-4
Ques: Simplify the expression by using the laws of exponents: 10-3 × 105? (2 marks)
Ans: According to exponent rules, when we have to multiply two expressions with the same base, we will add the exponents while the base of the exponents remains the same.
= This means, 10-3 × 105
= 10(-3 + 5)
= 102
= 100
Ques: What is the simplification of 93 × 92? (2 marks)
Ans: The solution is as follows:
= 93 × 92
= 93+2
Therefore the result is 95
Ques: Simplify and find the value of 162/42? (2 marks)
Ans: We can write the given expression as follows:
= 162/42
= (16/4)2
= 42
Therefore the result is 16
Ques: Find the value of (81)¾? (2 marks)
Ans: The solution is as follows:
= (81)3/4
= (34)3/4
= 34×(3/4)
= 33
Therefore the result is 27
Ques: Find the value of x if 64 = 16/4x? (3 marks)
Ans: We have 64 = 16/4x
⇒ 43 = 42/5x
⇒ 43 = 42-x
Since the quantity is the same on both sides and bases are also the same, hence, exponents will also be the same.
⇒ 3 = 2-x
⇒ x = 2-3
Therefore the result is -1
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