What is the standard form?

In mathematics, everything has a standard form. The standard form is like the short form of a particular function.

Suppose, there is an addition of the three numbers as $$348+5+9=362$$, so $$348+5+9$$ will be the expansion form, and $$362$$ is the standard form. Hence, it is derived that standard form is the simplified form of the expanded form.

There are  different standard forms, like the standard form of an equation, standard form of a polynomial and the standard linear equation. The standard form formula is  used to help in finding the general representation for the different types of notations. 

• The standard form of the line is $$y=ax$$. 

• The standard form of the parabola is $$y=ax^{2}$$

For example, when the whale jumps out of water for taking oxygen, they form the parabola $$y=ax^{2}$$ where $$a$$ can be any length.

E1.7A: Understand the meaning of indices (fractional, negative, and zero) and use the rules of indices. 

There are $$6$$ laws of  indices. They are multiplying, dividing, raising a power of a power , power of zero , fractional, and negative indices. 

Law 1

The first law is multiplying indices. If there is one variable multiplied by itself with a different power, the result will be the variable itself with some of the power. It is represented as $$p^a \times p^a =p^{a+b}$$. 

Law 2

The second law is dividing indices. If one variable is divided by itself with the different power,  the result will be the variable itself with the subtraction of the power. It is represented as $$p^{a} \div p^{b}=p^{a-b}$$. 

Law 3

The third law is raising a power to a power where the powers are multiplied to  each other. It is represented as $$({p^{a})}^{b}=p^{a \times b}$$. 

Law 4

The fourth law is the power of zero. The power is zero is always equal to one because any number divided by itself gives the quotient as $$1$$. It is represented as $$a^{0}=1$$.

A real-life example is a car garage that is fifteen feet by fifteen feet square. So, the total area will be $$15 \times 15$$ feet that are $$225$$ square feet.

Worked examples on indices

Example 1: If a square of length has $$5$$ cm each, find the area of the square.

Step 1: Area of the square.

The length of the square is $$r=5$$ cm and the area of the square is $$r^{2}$$.

Step 2: Substituting the values in the area of square formula.

$$5^{2}=25\text{cm}^{2}$$.

So, area of the square is $$25\text{cm}^{2}$$.

Example 2: Solve the equation $$5a^{6} \times 3a \times 2a^{3}$$.

Step 1: Solve the equation $$5a^{6}\times 3a$$.

Using multiplying indices while solving the equation as $$5a^{6} \times 3a=15a^{7}$$.

Step 2: Solve the other part of equation $$15a^{7} \times 2a^{3}$$.

$$15a^{7} \times 2a^{3}=30a^{10}$$

So, the solution of $$5a^{6} \times 3a \times 2a^{3}=30a^{10}$$.

E1.7B: Use the standard form $$A \times 10^{n}$$ where $$n$$ is a positive or negative integer, and $$1 \leqslant A < 10$$.

Law 5

The fifth law is negative indices. If a variable can be divided by itself by another degree, the result is the variable itself, minus its degree. It is represented as $$p^(-a)=\frac{1}{p^{a}}$$. The example is $$a^{5} \div a^{8}=a^{-3}$$ so it can also be written as $$\frac{1}{3}$$.

Law 6

The sixth law is fractional indices. If one variable is having a power in the fraction like $$x^{\frac{1}{n}}$$,  it can also be written as $$\sqrt[n]{x}$$. It is represented as $$x^{\frac{1}{n}} = \sqrt[n]{x}$$. It can also be represented as $$x^{\frac{m}{n}} = (\sqrt[n] {x})^{m}$$.

Worked examples on standard form

Example 1: Solve $$100^{\frac{3}{2}}$$.

Step 1: Convert the equation into the form $$(\sqrt[n]^{x})^{m}$$.

$$100^{\frac{3}{2}}=(\sqrt[2]{100})^{3}$$.

Step 2: Solve the square root of $$(\sqrt[2]{100})^{3}$$.

$$(\sqrt[2]{100})^{3}=10^{3}$$.

So, the solution of $$100^{\frac{3}{2}}=1000$$.

Example 2: Solve $$16^{-\frac{3}{2}}$$.

Step 1: Convert the equation into positive form.

$$16^{-\frac{3}{2}}=\frac{1}{16^{\frac{3}{2}}}$$.

Step 2: Convert the equation into the form $$(\sqrt[n]^{x})^{m}$$.

$$\frac{1}{16^{\frac{3}{2}}}=\frac{1}{(\sqrt[2] {16})^{3}}$$.

Step 3: Solve the square root of $$\frac{1}{(\sqrt[2] {16})^{3}}$$.

$$\frac{1}{(\sqrt[2] {16})^{3}}=\frac{1}{4^{3}}$$.

So, the solution of $$16^{-\frac{3}{2}}=\frac{1}{64}$$.