What are algebraic indices?

An index is used to show how many times a number is multiplied by itself. The plural word for index is indices. If some number is raised to some power, then the power it is raised to is the index of that number. For instance, $$2^{4}=2\times 2\times 2\times 2$$. So, $$2$$ is the base and $$4$$ is the index.

E2.4A: Use and interpret positive, negative and zero indices.

Positive indices: If the index of any number is a whole number, then it is termed as a positive index. For instance, $$2^4$$, $$a^5$$.

Negative indices: If the index of any number is negative, then it is termed as negative index. For instance, $$a^{-2}$$, $$b^{-5}$$. Here, $$a^{-m}$$ can be written as $$\frac{1}{a^{m}}$$.

Zero indices: If the index of any number is zero, then it is termed a zero index. For instance, $$5^0$$, $$x^0$$. The value of zero indices is always equal to $$1$$.

For solving indices, there are some rules. 

Worked examples of algebraic indices

Example 1: Determine if $$5^4\times 5^3$$ has a positive, negative or zero index.

Step 1: Recall the formula $$a^{m}\times a^{n}=a^{m+n}$$

Therefore, $$ 5^4\times5^3=5^{4+3}=5^7$$

Step 2: Determine the index type

Since the power is positive, it is a positive index.

Example 2: Determine if $$7^2\times 7^{-6}$$ has a positive, negative or zero index.

Step 1: Recall the formula $$a^{m}\times a^{n}=a^{m+n}$$

Therefore, $$7^2\times 7^{-6}=7^{-2+6}=7^{-4}$$

Step 2: Determine the index type

Since the power is negative, it is a negative index.

Example 3: Simplify $$\frac{\left(3x^2y^2\right)^2\times\left(4xy\right)^3}{24x^4y^7}$$.

Step 1: Simplify $$\left(3x^2y^2\right)^2$$

$$\left(3x^2y^2\right)^2=3\times3\times x^4y^4$$

Step 2: Simplify $$\left(4xy\right)^3}$$

$$\left(4xy\right)^3=4\times4\times4\times x^3y^3$$

Step 3: Use the rules of indices to simplify them further

$$\frac{\left(3x^2y^2\right)^2\times\left(4xy\right)^3}{24x^4y^7}$$

$$\frac{3\times3\times{x^4y^4}\times{4\times4\times4\times}x^3y^3}{24x^4y^7}$$

$$\frac{24x^{4+3}y^{4+3}}{x^4y^7}$$

$$24x^{7-4}y^{7-7}$$

$$24x^3y^0 = 24x^3$$

Thus, the simplified value of $$\frac{\left(3x^2y^2\right)^2\times\left(4xy\right)^3}{24x^4y^7}$$ is $$24x^3$$.

E2.4B: Use and interpret fractional indices.

Fractional indices: If the index of a number is in fractional form, then it is termed fractional indices. For instance, $$5^\frac{1}{2}$$, $$7^\frac{1}{4}$$, $$6^\frac{2}{5}$$. The numerator of the fraction is power, and the denominator of the fraction stands for the type of root. For instance, in $$10^{\frac{7}{9}}$$, $$9^{\text{th}}$$ the root of $$10$$ is multiplied $$7$$ times.

Worked examples of fractional indices

Example 1: Express $$\left(\sqrt[4]{8}\right )^6$$ in the form of $$a^\frac{m}{n}$$.

Step 1: Recall the formula $$\sqrt[n]{a}=a^\frac{1}{n}$$

$$\sqrt[4]{8}=8^\frac{1}{4}$$

Step 2: Recall the formula $$\left (a^m \right )^{n}=a^{mn}$$

$$\left (8^\frac{1}{4} \right )^{6}=8^{\frac{1}{4}\times6}$$.

Step 3: Simplify to get the simplest form of the fractional index

$$8^{\frac{1}{4}\times6}=8^{\frac{3}{2}}$$.

Example 2: Rewrite $$p^{\frac{6}{7}}$$ in the form of $$\left(\sqrt[n]{a}\right )^{m}$$.

Step 1: Factorise the index power

$$\frac{6}{7}=\frac{1}{7}\times6$$

Step 2: Substitute the value of $$\frac{6}{7}$$

$$p^{\frac{6}{7}}=p^{\frac{1}{7}\times6}$$

Step 3: Apply appropriate law of indices

$$p^{\frac{1}{7}\times6}=\left(p^\frac{1}{7} \right )^6$$

$$\left(\sqrt[7]{p} \right )^6$$

E2.4C: Use the rules of indices.

In this section, we learn how to use these rules of indices efficiently. 

Worked examples of the rule of indices

Example 1: Simplify $$3^4\times3^6$$.

Step 1: Recall the law of indices that has to be used

$$x^{m}\times x^{n}=x^{m+n}$$

Step 2: Apply formula and get the answer 

$$3^4\times3^6 =3^{4+6}$$

$$3^{10}$$

Thus, $$3^4\times3^6=3^{10}$$.

Example 2: Evaluate $$\frac{h^{-3}\times h^{6}}{h^5\times h^{-3}}$$.

Step 1: Recall the laws of indices that has to be used

$$x^{m}\times x^{n}=x^{m+n}$$

$$x^{m}\div x^{n}=x^{m-n}$$

Step 2: Simplify the numerator of the expression

$$h^{-3}\times h^{6}=h^{-3+6}$$

$$h^{3}$$

Step 3: Simplify the denominator of the expression

$$h^5\times h^{-3}=h^{-3+5}$$

$$h^2$$

Step 4: Simplify the resultant

$$\frac{h^3}{h^2}=h$$

Thus, $$\frac{h^{-3}\times h^{6}}{h^5\times h^{-3}}=h$$

Example 3: Evaluate $$4^{\left (\frac{3}{2} \right )^\frac{2}{3}}$$.

Step 1: Recall the laws of indices that have to be used

$$\left ( a^m \right )^{n}=a^{mn}$$

Step 2: Apply the formula and get the answer

$$4^{\left (\frac{3}{2} \right )^\frac{2}{3}} = 4^{\frac{3}{2}\times\frac{2}{3}}$$

$$4^{\frac{6}{6}}$$

$$4$$.

$$4^{\left (\frac{3}{2} \right )^\frac{2}{3}}=4$$