1.4 - Laws of Exponents (Indices)
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Product of Powers
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Today we are learning about the Product of Powers. When we multiply powers with the same base, we add the exponents. Can anyone give me an example?
Is it like when we say 2^3 * 2^2? We can add 3 and 2 to get 2^(3+2).
Exactly! So that means 2^3 * 2^2 = 2^5, which equals 32. To remember this, you can think 'adding exponents, when powers come together'—does that make sense?
Yes! It's like a party where powers invite exponents to join!
So if I multiplied 3^4 * 3^1, I would get 3^(4+1).
Exactly right! Now, what do we get if we simplify that?
That's 3^5, which is 243!
Great job! Let’s summarize: when multiplying powers with the same base, we add the exponents.
Quotient of Powers
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Next, we're going to look at the Quotient of Powers. Who can tell me what happens when we divide powers with the same base?
We subtract the exponents! Like a^m over a^n is a^(m-n).
Exactly! Can someone give an example?
Sure! If I have 5^6 over 5^3, it's 5^(6-3) which is 5^3.
Right! And what is 5^3 equal to?
That’s 125!
Fantastic! Remember, when dividing powers, it’s all about subtraction, just like 'take away with the quotients'!
Power of a Power
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Now, what happens when we raise a power to another power?
Oh, we multiply the exponents! Like (a^m)^n = a^(m*n).
Very good! Can anyone provide an example?
If we have (4^2)^3, we can multiply 2 and 3 to get 4^(2*3).
Correct! And what does that simplify to?
That’s 4^6, which equals 4096!
Great work! Just remember: when you raise a power, it's 'multiply the tops when the powerful crops'!
Zero and Negative Exponents
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We've learned about positive exponents now let's talk about zero and negative exponents. Who can tell me what any number raised to the power of zero equals?
That's easy! Anything to the power of zero equals one!
Exactly! And what does a^{-n} represent?
That means one over a^n, right?
Yes! That’s correct! If I say 2^{-3}, can someone tell me what that would be?
That would equal 1/2^3, which is 1/8.
Well done! So, for exponents, remember that 'zero means one, negatives are fun – flip it!'
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The laws of exponents cover essential mathematical rules that dictate how to handle algebraic operations involving indices. Key operations include multiplication and division of powers, exponentiation, and the use of zero and negative exponents.
Detailed
Laws of Exponents (Indices)
This section explains the fundamental laws of exponents relevant for any non-zero real number a and integers m and n. The key laws presented here are:
- Product of Powers: When multiplying two powers with the same base, add the exponents:
$$ a^m \cdot a^n = a^{m+n} $$
- Quotient of Powers: When dividing two powers with the same base, subtract the exponent in the denominator from the exponent in the numerator:
$$ \frac{a^m}{a^n} = a^{m-n} $$
- Power of a Power: When raising a power to another power, multiply the exponents:
$$ (a^m)^n = a^{mn} $$
- Zero Exponent: Any base raised to the power of zero is equal to one:
$$ a^0 = 1 $$
- Negative Exponent: A negative exponent indicates a reciprocal:
$$ a^{-n} = \frac{1}{a^n} $$
Understanding these laws is crucial for simplifying expressions in algebra and lays the foundation for more advanced mathematics in future studies.
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Product of Powers
Chapter 1 of 5
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Chapter Content
For any non-zero real number a and integers m, n:
● am⋅an=am+na^m ullet a^n = a^{m+n}
Detailed Explanation
The Product of Powers rule states that when you multiply two powers that have the same base, you add their exponents. For example, if you have a² and a³, you can multiply them as follows: a² × a³ = a^{2+3} = a^5. This means the powers contribute their values to a single new exponent, simplifying the expression.
Examples & Analogies
Imagine you are stacking boxes. If you have 2 boxes in one stack (a²) and 3 boxes in another (a³), when you combine them (multiply the stacks), you end up with a taller stack of 5 boxes (a^5).
Quotient of Powers
Chapter 2 of 5
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Chapter Content
● aman=am−n
\frac{a^m}{a^n} = a^{m-n}
Detailed Explanation
The Quotient of Powers rule indicates that if you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. For instance, a⁵ divided by a² can be simplified to a^{5-2} = a^3. This subtraction reflects the decreasing count of power when splitting.
Examples & Analogies
Think of a situation where you have 5 apples (a⁵) and you give away 2 apples (denominator a²). You will have 3 apples left (a³), illustrating the subtraction of the counts of each base.
Power of a Power
Chapter 3 of 5
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Chapter Content
● (am)n=amn
(a^m)^n = a^{mn}
Detailed Explanation
The Power of a Power rule explains that when you raise a power to another power, you multiply the exponents. For example, (a²)³ means you take a² and raise it to the power of 3, which simplifies to a^{2×3} = a^6. This multiplication highlights the compounded effect of raising powers multiple times.
Examples & Analogies
If you think of a recipe where you double a dish twice, each doubling is raising the amount you started with to a higher power. So, doubling something that is already doubled gives you four times the original!
Zero Exponent Rule
Chapter 4 of 5
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Chapter Content
● a0=1
a^0 = 1
Detailed Explanation
The Zero Exponent Rule states that any non-zero number raised to the power of zero equals one. This might seem counterintuitive, but it helps maintain consistency in mathematical rules. For instance, a² divided by a² is a^{2-2} = a^0, which equals 1.
Examples & Analogies
Consider how you might have zero dollars (a^0). Although you have no money, the idea is that you still represent having 'one unit' of currency as the concept of 'nothingness.'
Negative Exponent Rule
Chapter 5 of 5
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Chapter Content
● a−n=1/a
a^{-n} = \frac{1}{a^n}
Detailed Explanation
The Negative Exponent Rule indicates that a negative exponent signifies the reciprocal of that base raised to the opposite positive exponent. For example, a^{-3} means you take 1 divided by a^3, or the fraction 1/a³. This provides a way of writing division in terms of multiplication.
Examples & Analogies
If you owe a friend 3 apples (a^{-3}), you can think of it as giving them a negative amount. However, to balance it, you can consider it as the equivalent of having to repay 1/(a³) in terms of apples later on.
Key Concepts
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Product of Powers: When multiplying powers with the same base, add the exponents.
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Quotient of Powers: When dividing powers with the same base, subtract the exponents.
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Power of a Power: When raising a power to another power, multiply the exponents.
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Zero Exponent: Any non-zero base raised to the power of zero equals one.
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Negative Exponent: A base raised to a negative exponent equals the reciprocal of that base raised to the absolute value of the exponent.
Examples & Applications
Example of Product of Powers: 3^4 * 3^3 = 3^(4+3) = 3^7 = 2187.
Example of Quotient of Powers: 5^5 / 5^2 = 5^(5-2) = 5^3 = 125.
Example of Power of a Power: (2^3)^2 = 2^(3*2) = 2^6 = 64.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Multiply and add, for powers using the same; Divide and subtract, remember the game!
Stories
Once upon a time, in the land of Exponentia, whenever two powers joined, they would add their courts. But if they were dividing, oh no! They would take away from their chests!
Memory Tools
For exponents: 'Add in pairs when multiplying, Subtract when dividing, Multiply when stacking, One's the answer when zero's backing!'
Acronyms
Remember PIZZ
Product is 'add'
Inverse is 'subtract'
Zero is '1'
and Z negative means 'flip it'.
Flash Cards
Glossary
- Exponent
A mathematical notation indicating the number of times a quantity is multiplied by itself.
- Base
The number that is raised to a power.
- Power
An expression that consists of a base and an exponent.
- Negative Exponent
An exponent that represents the reciprocal of a number raised to the absolute value of that exponent.
- Zero Exponent
Any non-zero number raised to the power of zero, which is equal to one.
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