Indicies - Explanation, Examples and Laws of Indicies

Welcome back to my blog!

Today, we'll be looking at indicies. Indicies, also known as exponents or powers, helps us express products of factors and large numbers in a more compact and efficient way. For example, instead of writing $2 \times 2 \times 2 \times 2 \times 2$, we can simply write it as $2^5$. Here, $2$ is the base, and $5$ is the power or exponent, indicating that $2$ multiplies itself $5$ times.

Example 1

Evaluate the following:

a. $4^4 = 4 \times 4 \times 4 \times 4 = 256$

b. $5^3\times 2^2\times 3^3=5\times 5\times 5\times 2\times 2\times 3\times 3\times 3=13500$

c. $2^3 \times 3^1 \times 7^2 = 2 \times 2 \times 2 \times 3 \times 7 \times 7 = 1176$

                                       Negative Bases

So far, we’ve only considered positive bases raised to a power. Let’s briefly explore negative bases:

• $(-2{)}^1=-2$
• $(-2{)}^2=-2\times -2=4$
• $(-2)^3 = -2 \times -2 \times -2 = -8$
  
• $(-2)^4 = -2 \times -2 \times -2 \times -2 = 16$
  

We can see a pattern from this. A negative base raised to an odd exponent yields a negative number, while an even exponent gives a positive number.

Note: Eliminating the brackets can lead to negative values, even when raised to a positive exponent. For example, $-2^4 = -1 \times 2^4 = -1 \times 16 = -16$.  So, always remember to put negative numbers in brackets when raising it to a power.

Example 2:

Evaluate the following expressions involving negative bases:

a. $(-\mathrm{6)}{\mathrm{^4}}^{}=-6\times -\mathrm{6\times }-\mathrm{6\times }-6=1296$

b. $(-3{)}^5=-3\times -3\times -3\times -3\times -3=-243$

c. $-(-4{)}^5=-(-4\times -4\times -4\times -4\times -4)<br><br>$

$=-(-1024)=1024$

                              Laws of Indices

The laws of indices provide us with rules to simplify calculations or expressions with powers of the same base:

  

1. $a^m \times a^n = a^{m+n}$

2. When dividing numbers with the same base, we maintain the base and subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
   

3. When raising a power to a power, we maintain the base and multiply the exponents: $(a^m)^n = a^{mn}$
   

4. When a product is raised to a power, each member is raised to that power: $(ab)^n = a^n \times b^n$

5. When a fraction is raised to a power, each part(the numerator and denominator) is individually raised to that power: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Example 3:

 Evaluate the Following

a. $a^3 \times a^2 = a^{3+2} = a^5$

b. $m^4 \times m^5 = m^{4+5} = m^9$

c. $3^5/3^3=3^{5-3}=3^2=9$

d. $\frac{p^7}{p^3} = p^{7-3} = p^4$

e. $(3a)^2 = 3^2 \times a^2 = 9a^2$

f. $\frac{(2x)}{y}^3 = \frac{(2x)^3}{y^3} = \frac{2^3 x^3}{y^3} = \frac{8x^3}{y^3}$

g. $(x^4{)}^5=x^{4\times 5}=x^{20}$

Zero and Negative Indices

The zero index law states that any non-zero number raised to the power of zero is $1$:

$$a^0 = 1 \quad (a \neq 0)$$

For example, $2^{3-3} = 2^0 = 1$.

The negative index law tells us that $a^{-n} = \frac{1}{a^n}$. This shows that $a^{-n}$ and $a^n$ are reciprocals.

Using the negative index law, we can also say $(a\mathrm{/b}{)}^{-n}={\left(\frac{\mathrm{b}}{a\mathrm{<br>}}\right)}^n<br>$

Example 4: 

Evaluate the Following

a. $7^0 = 1$

b. $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

c. $3^{0 - 3 - 1} = 1 - \frac{1}{3} = \frac{2}{3}$

d. $\left(\frac{5}{3}\right)^{-2} = \left(\frac{3}{5}\right)^{2} = \frac{3^2}{5^2} = \frac{9}{25}$

Solve more examples to strengthen your skills and don't hesitate to ask questions about anything you don't understand. Happy learning!

                                         Assignment

1. Simplify the following:

   a. $(1½)^{-3}$
   b. $3^0 + 3^1 - 3^{-1}$
   c. $\frac{(xy)^3}{y^{-2}}$
   d. Write the following as powers of $2$, $\mathrm{3<br>}$$5$:
   i. $125$
   ii. $\frac{32}{81}$
   iii. $2.4$