**Cube**

All objects in the world are three-dimensional. It means that everything in this world has length, breadth, and height. So we can say 3-D figures are solid figures.

A cube is a solid figure which has its entire sides equal in measurements. The measurements of quantity in an object or the volume or capacity of an object is measured by cubic measurements such as cubic centimeter or cubic meter.

The measurements of quantity in an object or the volume or capacity of an object is measured by cubic measurements such as cubic centimeter or cubic meter.

**For example** Volume= Length X Breadth X Height

= 1cm X 1cm X 1cm

= 1 cubic centimeter

= cm^{3}

Similarly 1m X 1m X 1m = m^{3 }= 1 cubic meter

We know that z^{2}= z X z.

And z^{n}= z X z X z X z ……… n times, So z^{3}= z X z X z

And z^{3} is called as z to the power 3 or z raise to 3 or z cubed.

**For example**:

1^{3}= 1 X 1 X 1= 1

2^{3}= 2 X 2 X 2= 8

(-5)^{3}= -5 X -5 X -5= -125

**Prime factorization of perfect cube**:

As we know that a perfect cube or a cube number obtained by multiplying a number by itself three times.

For example find out Prime factorization of 12^{3}.

We know 12= 2 X 2 X 3

So 12^{3}=12X12X12=1728

=2X2X3X2X2X3X2X2X3

=2^{3}X2^{3}X3^{3}

Note: Each prime factor of a number appears three times in the prime factorization of a cube of the number.

**Prime factorization to make Perfect cube**:

We can also check with the help of prime factorization whether a number is a perfect cube or not? How can we make that number as a perfect cube?

**For example**: Find out 243 is a perfect cube or not? If it is not a perfect cube what we need to do to make it a perfect cube?

**Solution:**

**Find out the prime factor of 243**

243= 3X3X3X3X3

=3^{3}X3^{2}

This is not a perfect square because here once 3 is repeated three times after that it is repeated two times.

So if we multiply on both side with 3 then this number became perfect cube of 9

Like 243X3= 3^{3}X3^{2}X3

729= 3^{3}X3^{3}

729=9^{3}

**Explanation**:

**Shortcut to find out the cube of any number with an example:**

**Step 1**: Take ten place number as “an” and unit place number as “b”

**Step 2**: Now use formula (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}

**Step 3**: Now use b^{3} to find out the last digit of any number cube if you will get two digit number then ten place number add to 3ab^{2}

**Step 4**: solve 3ab^{2} for ten place digit and write down if again you will get two digit number then unit place number write down as ten places of the result and ten places number add to the 3a^{2}b.

**Step 5**: solve 3a^{2}b for ten places digit and write down if again you will get two digit number then unit place number write down as hundred places of the result and ten place number add to the a^{3}.

**Step 6**: now solve a3 and write down as thousand places.

Complete review:

**Cube roots**

As we know that 4^{3}=64. So we can say that 64 is a cube of 4. Otherwise, 4 is a cube root 64. It is written as ten as

**Q1: find out the cube root of 3375.**

**Solution**: Prime factors of 3375=5X5X5X3X3X3

= 5^{3}X3^{3}

=5X3

=15

**Explanation**: