We have sent an OTP to your contact. Please enter it below to verify.
Alert
Your message here...
Your notification message here...
For any questions or assistance regarding Customer Support, Sales Inquiries, Technical Support, or General Inquiries, our AI-powered team is here to help!
Listen to a student-teacher conversation explaining the topic in a relatable way.
Signup and Enroll to the course for listening the Audio Lesson
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.
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'!
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'!
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!'
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
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.
Dive deep into the subject with an immersive audiobook experience.
Signup and Enroll to the course for listening the Audio Book
For any non-zero real number a and integers m, n:
● am⋅an=am+na^m ullet a^n = a^{m+n}
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.
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).
● aman=am−n \frac{a^m}{a^n} = a^{m-n}
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.
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.
● (am)n=amn (a^m)^n = a^{mn}
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.
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!
● a0=1 a^0 = 1
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.
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.'
● a−n=1/a
a^{-n} = \frac{1}{a^n}
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.
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.
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
Product of Powers: When multiplying powers with the same base, add the exponents.
Quotient of Powers: When dividing powers with the same base, subtract the exponents.
Power of a Power: When raising a power to another power, multiply the exponents.
Zero Exponent: Any non-zero base raised to the power of zero equals one.
Negative Exponent: A base raised to a negative exponent equals the reciprocal of that base raised to the absolute value of the exponent.
See how the concepts apply in real-world scenarios to understand their practical implications.
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.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
Multiply and add, for powers using the same; Divide and subtract, remember the game!
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!
For exponents: 'Add in pairs when multiplying, Subtract when dividing, Multiply when stacking, One's the answer when zero's backing!'
Review key concepts with flashcards.
Term
What is the law for multiplying powers with the same base?
Definition
What does a^0 equal?
How do we handle negative exponents?
What is the power of a power rule?
Review the Definitions for terms.
Term: Exponent
Definition:
A mathematical notation indicating the number of times a quantity is multiplied by itself.
Term: Base
The number that is raised to a power.
Term: Power
An expression that consists of a base and an exponent.
Term: Negative Exponent
An exponent that represents the reciprocal of a number raised to the absolute value of that exponent.
Term: Zero Exponent
Any non-zero number raised to the power of zero, which is equal to one.
Flash Cards
Glossary of Terms