10.3 - Laws of Exponents
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Understanding the Product of Powers Law
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Today, we will talk about the Product of Powers law. Can anyone remind me what the law states?
It says that when you multiply two powers with the same base, you add the exponents.
Exactly! If we have a^m × a^n, we write it as a^(m+n). Let's say we have 2^3 × 2^2. What would that equal?
That would be 2^(3+2), which is 2^5.
Correct! And 2^5 equals 32. Now, does this rule work for negative exponents too?
I think it should! Like if we had 2^(-2) × 2^(-3)...
You're right! That would be 2^(-2 + -3) = 2^(-5). Great job!
To remember this, think 'Add when you multiply!' Now, could anyone summarize this law for me?
When multiplying powers with the same base, add the exponents!
Exploring the Quotient of Powers Law
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Moving on, let’s discuss the Quotient of Powers Law. Who can explain how this works?
When you divide two powers with the same base, you subtract the exponents!
Perfect! So, for example, if we have a^m ÷ a^n, it becomes a^(m-n). Let's try an example: what is 5^4 ÷ 5^2?
That would be 5^(4-2), which equals 5^2.
Right! And 5^2 is 25. Now, if we had negative exponents, such as 5^(-1) ÷ 5^(-3), what would we do?
We would do -1 - (-3), which gives us 5^(2).
Exactly! So remember, 'Subtract when you divide.' Can anyone explain how we can write this another way?
It can also be written as 1/(5^2) if the exponent is negative!
Understanding the Power of a Power Law
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Let’s move to the Power of a Power Law. Can anyone tell me what that means?
It means when you raise a power to another power, you multiply the exponents!
Correct! So we write this as (a^m)^n = a^(m*n). Let’s see an example with 3^2 raised to the power of 4.
That would be 3^(2*4) = 3^8.
Great! And what is 3^8?
That's 6561!
Well done! Now, can anyone summarize this law?
Multiply the exponents when you raise a power to a power!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section explains the basic laws of exponents, how they apply to negative numbers, and provides examples of simplifying exponents through various mathematical operations. It highlights the significance of exponent rules in expressing mathematical expressions clearly and concisely.
Detailed
Detailed Summary of Laws of Exponents
In this section, we explore the key laws of exponents, which are rules used in the manipulation and simplification of expressions involving exponents.
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Product of Powers Law: For any non-zero integer a, the product of powers states that:
a^m × a^n = a^(m+n). This rule applies even when m and n are negative integers, confirming that this law is universal across positive, negative, and zero exponents. -
Quotient of Powers Law: This law states that:
a^m ÷ a^n = a^(m-n). Thus, when dividing powers with the same base, we subtract the exponents regardless of their signs. -
Power of a Power Law: When raising a power to another power, the rule is:
(a^m)^n = a^(m*n). This rule also holds for negative and positive exponents. -
Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to one:
a^0 = 1. -
Negative Exponent Rule: A negative exponent indicates a reciprocal:
a^(-m) = 1/(a^m). This means we can rewrite the negative exponent as a positive exponent in the denominator.
The section provides several examples for each law, ensuring clarity in understanding how to simplify expressions that involve these rules. Additionally, exercises are included to reinforce understanding and application of these laws.
Example : Simplify
(i) \[ \left( \frac{4}{5} \right)^{-3} \left( \frac{2}{3} \right)^{2} \]
(ii) \[ \left( \frac{6}{7} \right)^{-4} \left( \frac{9}{10} \right)^{-2} \]
Solution:
(i) \[ \left( \frac{4}{5} \right)^{-3} = \frac{5^{3}}{4^{3}} = \frac{125}{64} \]
\[ \left( \frac{2}{3} \right)^{2} = \frac{4}{9} \]
\[ \frac{125}{64} \times \frac{4}{9} = \frac{500}{576} = \frac{125}{144} \]
(ii) \[ \left( \frac{6}{7} \right)^{-4} = \frac{7^{4}}{6^{4}} = \frac{2401}{1296} \]
\[ \left( \frac{9}{10} \right)^{-2} = \frac{10^{2}}{9^{2}} = \frac{100}{81} \]
\[ \frac{2401}{1296} \times \frac{100}{81} = \frac{240100}{104976} \]
In general, \[ \left( \frac{a}{b} \right)^{m} = \frac{b^{m}}{a^{m}} \]
New Question: Simplify
(i) \[ \left( \frac{3}{4} \right)^{-2} \left( \frac{5}{6} \right)^{3} \]
(ii) \[ \left( \frac{8}{9} \right)^{-5} \left( \frac{11}{12} \right)^{1} \]
Solution:
(i) \[ \left( \frac{3}{4} \right)^{-2} = \frac{4^{2}}{3^{2}} = \frac{16}{9} \]
\[ \left( \frac{5}{6} \right)^{3} = \frac{125}{216} \]
\[ \frac{16}{9} \times \frac{125}{216} = \frac{2000}{1944} = \frac{500}{486} \]
(ii) \[ \left( \frac{8}{9} \right)^{-5} = \frac{9^{5}}{8^{5}} = \frac{59049}{32768} \]
\[ \left( \frac{11}{12} \right)^{1} = \frac{11}{12} \]
\[ \frac{59049}{32768} \times \frac{11}{12} = \frac{649539}{393216} \]
In general, \[ \left( \frac{a}{b} \right)^{m} = \frac{b^{m}}{a^{m}} \]
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Introduction to Laws of Exponents
Chapter 1 of 6
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Chapter Content
We have learnt that for any non-zero integer a, am × an = am + n, where m and n are natural numbers. Does this law also hold if the exponents are negative? Let us explore.
Detailed Explanation
In this introduction, we establish a foundational property of exponents: when multiplying two numbers with the same base, you can simply add their exponents. This is represented by the formula: \( a^m \times a^n = a^{m+n} \). This chunk raises a question whether the same applies when dealing with negative exponents, which leads us to explore further.
Examples & Analogies
Think of exponents like a recipe. Just as you combine quantities of ingredients to get the final dish, the exponent allows us to combine quantities of the same base number.
Applying the Law with Negative Exponents
Chapter 2 of 6
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Chapter Content
(i) We know that 2 – 3 = a−m for any non-zero integer a. Therefore, 2−3 ×2−2 = × = = = 2 – 5.
Detailed Explanation
Here, we explore how the law works with negative exponents. If we take \( 2^{-3} \) and \( 2^{-2} \) and multiply them, we use the rule of adding the exponents: \( 2^{-3} \times 2^{-2} = 2^{-3 + (-2)} = 2^{-5} \). This shows that the law holds true even when the exponents are negative.
Examples & Analogies
Imagine if you’re playing a game where gaining points is like using positive exponents, and losing points is like using negative ones. If you gain 3 points and then lose 2, your total change is similar to saying you have lost a net of 5 points.
Examples of Laws with Negative Bases
Chapter 3 of 6
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Chapter Content
(ii) Take (–3)– 4 × (–3)–3 = (−3)4 + (−3) = (−3)–7.
Detailed Explanation
This example demonstrates the application of the exponent rule to a negative base. Here, if you multiply \( (-3)^{-4} \) and \( (-3)^{-3} \), you add the exponents. Thus, it simplifies to \( (-3)^{-7} \). This confirms our initial rule of exponent addition still applies, regardless of the base being positive or negative.
Examples & Analogies
Think of a thermometer: moving above zero temperatures is like positive exponents, while moving below zero is like negative exponents. Combining both can give you a clearer picture of temperature changes, similar to adding exponents.
General Law of Exponents
Chapter 4 of 6
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Chapter Content
In general, we can say that for any non-zero integer a, am × an = am + n, where m and n are integers.
Detailed Explanation
This statement summarizes the law of exponents for all integers, confirming it works for both natural and negative integers. The formula emphasizes that when multiplying powers with the same base, the exponents can be added, which is crucial for simplification in mathematics.
Examples & Analogies
Think of adding weights: if you have weights of 3 kg and 2 kg, when combined they give you a total of 5 kg. Similarly, with exponents, combining them gives a total exponent.
Verifying Laws of Exponents
Chapter 5 of 6
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Chapter Content
On the same lines, you can verify the following laws of exponents, where a and b are non-zero integers and m, n are any integers.
Detailed Explanation
This chunk mentions additional laws of exponents which can be verified using previous principles. For example, the laws state that dividing powers with the same base involves subtracting the exponents, and raising a power to another power means multiplying the exponents.
Examples & Analogies
Consider laws of exponents as traffic rules: they help streamline calculations, simplifying complex processes into straightforward ones, just like following traffic rules makes travel safer and smoother.
Examples and Applications
Chapter 6 of 6
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Chapter Content
Let us solve some examples using the above Laws of Exponents.
Detailed Explanation
In this part, we can see specific problems and solutions that illustrate the laws of exponents in action. Each example not only applies the laws but also demonstrates how they simplify expressions.
Examples & Analogies
Think of solving problems in math as navigating through a maze. Each law is like a map that helps you find the quickest route to the solution, making complex tasks manageable.
Key Concepts
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Product of Powers Law: a^m × a^n = a^(m+n)
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Quotient of Powers Law: a^m ÷ a^n = a^(m-n)
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Power of a Power Law: (a^m)^n = a^(m*n)
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Zero Exponent Rule: a^0 = 1 for any non-zero a
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Negative Exponent Rule: a^(-m) = 1/(a^m)
Examples & Applications
Example: 2^3 × 2^2 = 2^(3+2) = 2^5 = 32
Example: 5^4 ÷ 5^2 = 5^(4-2) = 5^2 = 25
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When you multiply, exponents add, follow this rule, and don't be sad!
Stories
Imagine a fruit basket where each fruit type represents a base, and every fruit you add increases your collection's total, just like adding exponents when multiplying!
Memory Tools
Remember: 'Add when you multiply, subtract when you divide!'.
Acronyms
PAWS
Product
Add; Quotient
Subtract; Power
Multiply.
Flash Cards
Glossary
- Exponent
A number that indicates how many times to multiply the base by itself.
- Base
The number that is being multiplied in an expression with an exponent.
- Negative Exponent
An exponent that signifies the reciprocal of the base raised to the absolute value of the exponent.
- Zero Exponent
Any non-zero base raised to the power of zero equals one.
Reference links
Supplementary resources to enhance your learning experience.