Speed Notes
Synopsis For Quick Revision.
Topic At A Glance:
Squares and Square Roots for CBSE Class 8
In this blog post, we will learn about squares and square roots, which are important concepts in mathematics. We will see how to find the square of a number, how to find the square root of a number, and some properties and applications of squares and square roots.
What is a square of a number?
The square of a number is the result of multiplying the number by itself. For example, the square of 5 is 5 x 5 = 25. We can write this as $$5^2 = 25$$, where the exponent 2 indicates that we are multiplying 5 by itself twice. The symbol $$^2$$ is called the square symbol.
We can find the square of any positive or negative number. For example, the square of -3 is $$(-3)^2 = (-3) × (-3) = 9$$. Note that the square of a negative number is always positive.
Some numbers have special names when they are squared. For example, the square of 10 is 100, which is called a perfect square. A perfect square is a number that can be expressed as the product of two equal factors. For example, $$100 = 10 × 10$$, so 100 is a perfect square. Similarly, $$64 = 8 × 8$$, so 64 is also a perfect square.
What is a square root of a number?
The square root of a number is the inverse operation of squaring a number. That is, if we know the square of a number, we can find the original number by taking its square root.
For example, if we know that $$5^2 = 25$$, we can find that $$\sqrt{25} = 5$$. The symbol $\sqrt{}$ is called the radical sign or the square root sign.
We can also write this as $$25^{1/2} = 5$$, where the exponent $$1/2$$ indicates that we are taking the half power or the square root of 25.
The symbol $^{1/2}$ is called the square root symbol.
We can find the square root of any positive number.
For example, the square root of 81 is $$\sqrt{81} = 9$$.
Note: The square root of a positive number can be positive or negative.
For example, $$\sqrt{9} = \pm 3$$, because both 3 and -3 are valid solutions to the equation $$x^2 = 9$$.
However, by convention, we usually take the positive value as the principal square root.
We cannot find the square root of a negative number using real numbers. For example, there is no real number x that satisfies $$x^2 = -4$$. To deal with such cases, we need to use complex numbers, which are beyond the scope of this blog post.
Some numbers have special names when they are square rooted.
For example, the square root of 100 is 10, which is called a perfect square root. A perfect square root is a number that is the square root of a perfect square. For example, $$\sqrt{64} = 8$$, so 8 is a perfect square root.
Properties and applications of squares and square roots
There are many properties and applications of squares and square roots in mathematics and other fields. Here are some examples:
The area of a square with side length x is equal to $$x^2$$.
For example, if a square has a side length of 6 cm, its area is $$6^2 = 36 cm^2$$.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle with sides a and b and hypotenuse c, $$a^2 + b^2 = c^2$$.
For example, if a right triangle has sides of length 3 cm and 4 cm, its hypotenuse has length $$\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 cm$$.
The distance formula states that the distance between two points (x1,y1) and (x2,y2) in a coordinate plane is equal to $$\sqrt{(x2 – x1)^2 + (y2 – y1)^2}$$.
For example, if we want to find the distance between (1,-2) and (4,-6), we can use the formula as follows: $$\sqrt{(4 – 1)^2 + (-6 – (-2))^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 units$$.
The quadratic formula states that the solutions of a quadratic equation of the form $$ax^2 + bx + c = 0$$ are given by $$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$.
For example, if we want to solve the equation $$x^2 – 5x + 6 = 0$$, we can use the formula as follows:
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 – 4(1)(6)}}{2(1)}$$
$$= \frac{5 \pm \sqrt{25 – 24}}{2}$$
$$= \frac{5 \pm \sqrt{1}}{2}$$
$$= \frac{5 \pm 1}{2} = 3, 2$$.
These are some of the basic concepts and applications of squares and square roots.
We hope you enjoyed this blog post and learned something new. If you have any questions or feedback, please leave a comment below. Thank you for reading!
Hello, dear readers! Welcome to another exciting edition of Math Fun Time, where we explore the wonders of mathematics in a fun and engaging way. Today, we are going to talk about divisibility rules, which are handy shortcuts to check if a number can be divided by another number without doing long division. Sounds cool, right? Let’s dive in!
You probably already know some basic divisibility rules, such as:
- A number is divisible by 2 if it ends with an even digit (0, 2, 4, 6, or 8).
- A number is divisible by 5 if it ends with a 0 or a 5.
- A number is divisible by 10 if it ends with a 0.
But did you know that there are divisibility rules for other numbers too? For example:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 4 if the last two digits form a number that is divisible by 4.
- A number is divisible by 6 if it is divisible by both 2 and 3.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
These rules are easy to remember and apply, but what about some harder numbers, like 7, 11, or 13? Don’t worry, there are divisibility rules for them too, but they are a bit more complicated. Here they are:
- A number is divisible by 7 if you double the last digit and subtract it from the rest of the number, and the result is divisible by 7.
- A number is divisible by 11 if you alternately add and subtract the digits from left to right, and the result is divisible by 11.
- A number is divisible by 13 if you add four times the last digit to the rest of the number, and the result is divisible by 13.
Let’s see some examples of how these rules work:
- Is 539 divisible by 7? To check, we double the last digit (9) and get 18. Then we subtract it from the rest of the number (53) and get 35. Is 35 divisible by 7? Yes, it is! So 539 is also divisible by 7.
- Is 4629 divisible by 11? To check, we alternately add and subtract the digits from left to right: $$4 -6 +2 -9 = -9$$. Is -9 divisible by 11? Yes, it is! So 4629 is also divisible by 11.
- Is 1694 divisible by 13? To check, we add four times the last digit (4) to the rest of the number (169) and get $$169 +16 =185$$. Is 185 divisible by 13? Yes, it is! So 1694 is also divisible by 13.
Wow, these rules are amazing! They can save us a lot of time and effort when we need to check for divisibility. But how do they work? Why do they work? What is the logic behind them? Well, that’s a topic for another blog post. Stay tuned for more Math Fun Time!
I hope you enjoyed this blog post and learned something new. If you have any questions or comments, feel free to leave them below. And don’t forget to share this post with your friends who love math as much as you do!
Until next time,
Math Fun Time
