Cambridge O Level Mathematics (Syllabus D)/Squares, Square Roots, Cubes, Cube Roots

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If you have come across this concept for the first time, you might think that this topic is all about the study of squares and cubes. You might even be asking yourself, "Since when did squares and cubes are able to grow roots?"

No, they don't grow roots! Furthurmore, we are not talking geometry here. What this topic covers is square numbers and cubic numbers, fundamental concepts in algebra.

Squares[edit]

Squaring a number is, in general, the process of multiplying a number by itself. For example, the square of 2 is the process of multiplying 2 by 2, which makes 4. This is denoted by the superscript 2.

This means that: 2^2 = 2 \times 2 = 4

22 is read as the square of 2, 2 to the power of 2 (the latter '2' defining the square) or simply just 2 squared.

The process of squaring is a many-to-one function. Recall from the previous topic that a many-to-one function is a function where different inputs can lead to one similar output.

Why is the square function a many-to-one function? Don't forget that you have negative numbers as well. Squaring a negative number leads to the same result as squaring its positive counterpart.

\begin{align}
&3^2 = 3 \times 3 = 9 \\
&(-3)^2 = (-3) \times (-3) = 9
\end{align}

Notice that the result always ends up positive. This is because when you multiply two negative numbers together, they always end up positive as the negative numbers cancel out each other. Therefore you can say that a number which has been squared is always non-negative.

The number '2' that denotes the process of squaring a number is called an exponent, i.e. the exponent used for squaring numbers is 2. It basically means how many times the same number is multiplied again. So, if it is exponent 1, then it is just the original number (e.g. 91=9) while if it is to the exponent 2 (i.e. squaring), then it is multiplied by itself (e.g. 92=81).

An integer that is the result of another integer that is squared is called a square number or a perfect square. 4 is a perfect square because 22=4. 9, 25, 81 and 144 are other examples of perfect squares. (the square of 3, 5, 8 and 12, respectively)

Cubes[edit]

The process of cubing is similar to squaring, only that the number is multiplied three times instead of two. The exponent used for cubes is 3, which is also denoted by the superscript 3.

\begin{align}
&4^3 = 4 \times 4 \times 4 = 64 \\
&8^3 = 8 \times 8 \times 8 = 512
\end{align}

The cubic function is a one-to-one function. Why is this so?

This is because cubing a negative number results in an answer different to that of cubing it's positive counterpart. This is because when three negative numbers are multiplied together, two of the negatives are cancelled but one remains, so the result is also negative.

\begin{align}
&7^3 = 7 \times 7 \times 7 = 343 \\
&(-7)^3 = (-7) \times (-7) \times (-7) = -343
\end{align}

In the same way as a perfect square, a perfect cube or cube number is an integer that results from cubing another integer. 343 and -343 are examples of perfect cubes.

Square Roots and Cube Roots[edit]

If we multiply a number itself twice is called a square of that number. Example:- m²= m*m. In cube root we have to multiply a number trice by itself the resulting number is called the cube number. Example:- the cube of 2= 2*2*2= 8.

The invention of cube root is done by a great Indian mathematician Aryabhata in 499CE.

Furthur Exponentation[edit]

Common Square and Cube Numbers, and Numbers To Other Powers[edit]

Summary[edit]

Examination Tips[edit]

Past Paper Questions[edit]

EXAM TIPS In cube roots there are many application based question buy a book of cube root and practice it well. First cool your mind then understand the question properly because in application based question of cube root normally everybody get confused what the question is about practice a lot cool your mind do it step by step practice the formulas learn the steps. then you find no problem in your questions. Again i am telling understand the questions first with cool mind. Follow it you will find it helpful.