Math Antics - Exponents and Square Roots

Summary

This Math Antics video explores the concept of roots as the inverse operation of exponents. It explains how roots 'undo' exponents, using examples like finding the base of 4 squared (16) through square and cube roots. The video clarifies the difference between the root (or radical) sign and division sign, and demonstrates how to calculate roots, including square and cube roots, with a calculator. It emphasizes that while roots can be complex, focusing on perfect squares makes them more manageable.

Takeaways

• 🔢 Exponents and roots are inverse operations in mathematics.
• ➗ Roots are used to undo exponents, similar to how subtraction undoes addition and division undoes multiplication.
• 🌳 The term 'root' is used because it represents the base of a number, akin to the base of a tree.
• 📐 The root symbol, also known as the radical sign, is distinct from the division sign and is used to denote roots.
• 🔑 The index number above the root symbol indicates the number of times the base should be multiplied to return to the original number.
• 🤔 Finding roots can be complex and often requires a calculator or special algorithms for non-perfect squares.
• 🔄 The square root (2nd root) and cube root (3rd root) are the most common types of roots and have special names.
• 📉 Perfect squares have whole number square roots, which are easily found using multiplication tables.
• 📚 The script encourages practicing math to improve, emphasizing that most homework problems will involve easy roots with whole number answers.
• 🌐 For further learning, the video directs viewers to www.mathantics.com.

Q & A

What are the inverse operations of exponents?

The inverse operations of exponents are called 'roots'.

What is the relationship between addition and subtraction in terms of inverse operations?

Addition and subtraction are inverse operations because subtraction undoes addition.

How does the concept of a 'base' relate to roots?

The 'base' in a root operation is the number that, when raised to the power indicated by the root, gives the number under the root sign.

What is the term for the special math symbol used in root operations?

The special math symbol used in root operations is called the 'radical' sign.

How is the radical sign different from the division sign?

The radical sign is different from the division sign because it has a check mark-like front shape instead of a curved front.

What does the number above the radical sign represent?

The number above the radical sign represents the original exponent, indicating how many times the base should be multiplied.

What is the square root of 16 and how is it found?

The square root of 16 is 4, found by determining what number multiplied by itself (2 times) equals 16.

What is the cube root of 16 and how is it calculated?

The cube root of 16 is 2.51984... (approximately 2.52 when rounded to two decimal places), found by determining what number multiplied by itself three times equals 16.

Why are square roots and cube roots given special names?

Square roots and cube roots are given special names because they are the most common types of roots encountered in math.

What are 'perfect squares' and why are their square roots easy to find?

'Perfect squares' are the results of squaring whole numbers, and their square roots are easy to find because they result in whole numbers.

How can one check if the calculated cube root of 16 is correct?

One can check if the calculated cube root of 16 is correct by cubing the result (raising it to the power of 3) and seeing if it equals approximately 16.

Outlines

00:00

📚 Introduction to Roots

The script introduces the concept of roots as the inverse operation of exponents. It explains that exponents and roots are inverse operations that undo each other, using the example of squaring a number (4^2 = 16) and then finding the root to return to the original base (4). The term 'root' is justified by its similarity to 'base', which is the starting point of an exponent. The script then introduces the radical symbol, explaining its difference from the division symbol and how it is used to denote roots. It also demonstrates how to find the square root and fourth root of 16, illustrating the process of finding the base of an exponent.

05:08

🔍 Roots Calculation Challenges

This paragraph discusses the difficulty of calculating roots, particularly when they do not result in whole numbers. It uses the example of the cube root of 16 to illustrate that roots are not the same as division and cannot be easily calculated without a calculator or special algorithms. The script then shows how to use a calculator to find the cube root of 16, resulting in a complex decimal number. It emphasizes that while roots are often complex, most homework problems will involve easy roots with whole number answers. The paragraph also explains the special names for the square root (2nd root) and cube root (3rd root), and notes that square roots are the most common and often the easiest to find, especially for perfect squares.

10:09

📈 Conclusion on Exponents and Roots

The final paragraph summarizes the relationship between exponents and roots as inverse operations and highlights the importance of understanding the most common roots, the square root and cube root. It acknowledges the difficulty of finding roots but encourages learning perfect squares as an easy starting point. The script concludes by emphasizing the importance of practice and directs viewers to the Math Antics website for more information.

Keywords

💡Exponents

Exponents, also known as indices, are a mathematical notation indicating the number of times a base number is multiplied by itself. In the script, exponents are introduced as a type of operation that can be undone by roots. For example, '4 squared' (written as 4^2) means multiplying 4 by itself once, resulting in 16. Exponents are foundational to understanding the relationship between multiplication and repeated multiplication.

💡Inverse Operations

Inverse operations are pairs of mathematical operations that can reverse each other's effect. The script explains that addition and subtraction, as well as multiplication and division, are inverse operations. In the context of exponents and roots, roots are the inverse operations that can undo the effect of exponents, helping to revert a number raised to a power back to its base form.

💡Roots

Roots are the inverse operations of exponents. They are used to find the base of a number that has been raised to a certain power. The script uses the example of finding the square root of 16 (which is 4, since 4^2 = 16) to illustrate how roots 'undo' the operation of exponents. Roots are crucial for understanding the concept of undoing exponentiation.

💡Base

The base in an exponentiation operation is the number that is being raised to a power. The script mentions that when performing a root operation, the goal is to find the base of the original exponent. For instance, in the expression √16 (the fourth root of 16), the base that when multiplied by itself four times gives 16 is 2.

💡Radical Sign

The radical sign, represented by the symbol √, is used to denote the root of a number. The script humorously refers to it as looking 'radical' and resembling a check mark, distinguishing it from the division sign. It is used to indicate that a root operation is being performed, as in √16, which asks what number multiplied by itself a certain number of times equals 16.

💡Square Root

The square root is a specific type of root, where the operation is finding a number that, when multiplied by itself once, equals the original number. The script explains that the square root is so common that it is the default root operation, indicated by the radical symbol without a number above it, as in √16, which means finding the number that when squared equals 16.

💡Cube Root

The cube root is another common type of root operation, where the goal is to find a number that, when multiplied by itself twice (three times in total), equals the original number. The script uses the cube root of 16 as an example, explaining that the cube root of 16 is 2, since 2^3 = 8 and not 16, but if we consider four 2's multiplied together (2^4), we get 16.

💡Perfect Squares

Perfect squares are the results of squaring whole numbers, such as 1, 4, 9, 16, etc. The script points out that finding the square root of perfect squares is straightforward because the answers are whole numbers, which are easily found using multiplication tables. This concept is important for understanding when roots can be easily calculated without a calculator.

💡Multiplication Table

The multiplication table is a learning tool used to memorize the products of multiplying numbers, typically from 1 to 12. The script uses the multiplication table to explain how perfect squares are easily identified and their square roots easily found, as they are located on the diagonal of the multiplication table.

💡Algorithms

Algorithms are step-by-step procedures for calculations. The script mentions that there are special algorithms for calculating roots, but they are complicated and thus are saved for future discussion. Algorithms are essential for understanding how to compute roots accurately, especially when they cannot be easily determined.

Highlights

Exponents have inverse operations called 'roots'.

Inverse operations are pairs of math operations that undo each other.

Exponentiation involves multiplying a number by itself a certain number of times.

Roots are used to find the base of an exponentiation.

The term 'root' is analogous to the base of a tree, which helps in remembering its function.

Root operations start with the result of an exponentiation to find the original base.

The root symbol, or radical sign, is used to denote roots.

The radical sign is distinct from the division sign by its check mark-like front shape.

The number under the root symbol is the one to take the root of.

The root symbol includes the original exponent, indicating how many times to multiply.

The 2nd root of 16 is 4, and the 4th root of 16 is 2.

Calculating roots can be complex and often requires a calculator or special algorithms.

Roots are not the same as division, even though the symbols look similar.

The square root and cube root are the most common types of roots.

The square root is the default root when no index number is specified.

Perfect squares have whole number square roots that are easy to find.

Learning perfect squares is a good starting point for understanding roots.

Most roots encountered in homework or tests will have nice whole number answers.

Practicing math problems is essential for getting better at understanding exponents and roots.

Transcripts

Hi! Welcome to Math Antics.

In our last video called “Intro to Exponents”, we learned that exponents (also called Indices) are a special type of math operation.

In this video, we’re going to expand on what we know about exponents

by learning about their inverse operations, which are called “roots”.

That’s kind of a strange name for a math operation, but it will make more sense in just a minute.

First, let’s review what we mean by inverse operations.

In the video called “What Is Arithmetic”, we learned that inverse operations are pairs of math operations that UNDO each other.

For example, you can undo addition by doing subtraction, so addition and subtraction are inverse operations.

Likewise, you can undo multiplication by doing division, so multiplication and division are inverse operations.

As I mentioned, exponents have inverse operations also.

There are operations that can undo them, and those operations are called roots.

To see how roots and exponents work together to undo each other,

let’s look at the simple exponent 4 to the 2nd power (or four squared).

Previously, we learned that this is the same as 4 × 4 which equals 16.

Doing this exponent meant going from 4 squared to 16.

So now if we want to undo that with a root operation,

that involves starting out with 16 and then somehow getting back to the 4 which is being raised to the 2nd power.

And do you remember what that part of the original exponent is called?

Yep, it’s called the “base”.

So doing a root operation is going to give us the base as our answer, and that helps us understand its name a little better.

The words “base” and “root” have a similar meaning, especially if you think of a tree.

The root is at the base of a tree, and that can help you remember how root operations work.

With a root operations, you start with the answer of an exponents and try to figure out what the base of that original exponent is.

Okay, but how do we actually do that?

How do we use a root operation to go backwards and figure out the base of the original exponent?

Well for starters, we need to know about a special math symbol that looks like this.

And you guessed it… it’s called the root sign.

Whoa Dude!

That math symbol looks totally radical dude!

It’s… it’s like that division thingy, only way cooler!!

Ah yes, that reminds me… the root sign is often referred to as the “radical” sign

(and mathematicians used that term even before surfers did)

And yes, it does look similar to the division sign so it’s really important not to get them confused.

The root (or radical) sign is different from the division sign because,

instead of having a curved front, its front shape is like a check mark.

The number that you want to take the root of goes under the sign like this.

So when you see a number under a root (or radical) sign like this, you know you need to figure out the base of the original exponent.

In this case, you need to figure out what number you could multiply together a certain number of times to get 16.

Ah - but there’s the catch! How many times?

The answer we get from taking the root will depend on how many times that number would be multiplied together.

But that would depend on the original exponent. So how do we know what that number is?

Simple… the root symbol tells us.

The root symbol actually includes the original exponent in it.

What? You don’t see it?

Oh… That’s because I didn’t draw it yet. And later in this video, you’ll understand why.

So let’s put a little 2 right here above the check mark part of the root symbol.

And that 2 tells us that we need to figure out what number (or base) could be multiplied together 2 times in order to get 16.

And, if you remember your multiplication table (or if you just look at our original example here)

you’ll know that the answer to that is 4.

Now do you see how the root operation is the inverse of the exponent operation?

When doing the exponent, we asked, “What do we get if we multiply 4 together 2 times?”, and the answer was 16.

But when we did the root operation, we asked, “What number could we multiply together 2 times to get 16?” and the answer was 4.

Great! Now that you understand how exponents and roots are related, we’re going to look closer at how root operations work.

To do that, we’re gonna change our root problem slightly. Let’s change the little 2 into a little 4.

The first root was asking us to figure out what number we could multiply together 2 times to get 16.

But this new root is asking us to figure out what number we could multiply together 4 times to get 16.

That’s a bit trickier, huh? Can you think of a number like that?

Yep, the answer is 2, because if you multiplied four ‘2’s together (2 × 2 × 2 × 2) you get 16.

So the 2nd root of 16 is 4, but the 4th root of 16 is 2.

Both those roots were pretty easy to figure out, right?

But unfortunately, figuring out roots in math can be much harder.

For example, what if we had this problem instead, “root 3 of 16”

That means we need to figure out what number we could multiply together 3 times to get 16.

Can you think of a number like that?

[No] I can’t either!

And unfortunately, it’s not easy to calculate what that number would be.

Remember, even hough this look a little but like the division symbol, this is NOT just division!

You can’t just divide 16 by 3 to get the answer. Roots are NOT the same as long division.

So how DO we calculate a root like this?

Well, there are special algorithms that you can use to calculate just about any root,

but they’re kinda complicated, so we’ll save those for a future video.

Instead, I’m gonna use a special root function on my calculator to get the answer.

And on my calculator the button for that root function looks like this.

To use it, I first enter the number that I want to take the root of, which is 16.

Next, I hit the root function button, and then I enter 3 so it knows that I want the 3rd root of 16.

Last, I hit the equals sign and voila… the answer is 2.519842… and the decimal digits just keep on going forever.

Wow! See what I mean about roots being hard to figure out?

That is a really complicated decimal number and you may even wonder if it’s the right answer. Well, let’s check…

Based on what we know about exponents and roots, if we multiply this decimal number together 3 times, we should get 16, right?

But to make it easier to check, let’s just round the number off to 2 decimal places. Let’s make it 2.52.

If we multiply 2.52 together 3 times, (in other words if we take 2.52 to the 3rd power) we’ll get: 16.003.

Well… that’s almost right. It’s really close to 16, isn’t it?

The reason it’s not exactly 16 is that we rounded the number off which made it less accurate.

But the more decimal digits we use, the closer we’ll get to 16.

In math, the vast majority of roots are complicated number like this.

And they’re hard to figure out unless you use a calculator or a special algorithm.

That’s the bad news.

But the good news is that most of the time, the roots you’ll be asked to do in your homework or on tests are the easy ones;

...the ones that have nice whole number answers.

And usually, you’ll only be asked to find 2nd or 3rd roots of numbers.

Do you remember in the last video, we learned that 2 and 3 are the most common exponents.

…so common in fact, that they even had special names.

Raising a number to the 2nd power was called “squaring” it,

and raising a number to the 3rd power was called “cubing” it.

Well, it’s the same with roots. Since the roots 2 and 3 are the most common, they get special names also.

The 2nd root is called the “square root”

and the 3rd root is called the “cube root”.

In fact, the “square root” is SO common that it’s basically the default root and its symbol even gets special treatment.

Do you remember that when I first showed you the root symbol, I left out the index number that tells you what root to find?

Well, whenever that number is left out, you can just assume that it’s 2.

In other words, the root symbol, with no index number, is always the square root.

So if you want someone to find a different root (like cubed or 4th or 5th)

then you need to include that number so they know which root to find.

And even though square roots are the most common, they’re not always easy to find.

Most are still going to be big long decimal numbers, except for the “perfect squares”.

It’s easier to find the square roots of the perfect squares because their answers can be found using the multiplication table.

On the multiplication table, have you ever noticed that all the answers to problems

where the same number is being multiplied together are on the diagonal of the table.

In other words, 2x2=4,

3 × 3 = 9,

4 × 4 = 16,

5 × 5 = 25,

6 × 6 = 36, and so on.

Well, those numbers are called the perfect squares because they’re the answers you get when you square a whole number.

And that means if you take the square root of a number along that diagonal, you get a nice whole number as your answer.

The square root of 4 is 2.

The square root of 9 is 3.

The square root of 16 is 4.

The square root of 25 is 5, and so on.

See what I mean?

Those roots are really common and they’re also easy to figure out if you know your multiplication facts.

So if you’re new to exponents and roots, learning the perfect squares is the place to start.

Once you understand how those exponents and roots work, you’ll be ready to figure our tougher problems.

Alright… so now you know how exponents are related to roots. They’re inverse operations and they undo one another.

And you also know that, just like 2 and 3 are the most common exponents,

the square root and the cube root are the most common roots.

You also know that finding roots is usually not very easy.

That’s important to know so you don’t get discouraged if you feel like it’s hard to figure out what a certain root is.

You’re not alone! We think it’s hard too and would normally just use a calculator to find them.

The good news is that some roots are easy to find, like the perfect squares, so be sure to focus on learning them first.

And remember, to get good at math you need to actually practice what you learn from watching videos,

so be sure to do some exercise problem.

And as always, thanks for watching Math Antics and I’ll see ya next time.

Learn more at www.mathantics.com