Laws Of Indices | A Level Maths Revision Notes

Contents hide

1 LEARNING OBJECTIVES:

2 MATH CONCEPTS

3 DISCUSSION

4 DISCUSSION

LEARNING OBJECTIVES:

To define index and base of a number
To discuss the different laws of indices
To evaluate expressions using the laws of indices

MATH CONCEPTS

INDEX OF A NUMBER- The index of a number tells how many times to use the number in a multiplication. It is written as a small number to the right and above the base number. It is also known as exponent or power.
BASE OF A NUMBER- It is the number that gets multiplied when using an index or exponent.
THE THREE BASIC LAWS OF INDICES For any real number x, y, m, and n, the following rules uphold:
- Multiplication Law/ Product Rule $x^{m}\cdot x^{n}=x^{m+n}$
- Division Law/Quotient Rule $x^{m}/x^{n}=\frac{x^{m}}{x^{n}}=x^{m-n}$
- Power Law/Power Rule $(x^{m})^{n}=x^{m\cdot n}$
Other Laws:
- $x^{-n} = \frac{1}{x^{n}}$
- $x^{\frac{m}{n}}=\sqrt[n]{x^{m}}$
- $(xy)^{m}=x^{m}y^{m}$
- $(\frac{x}{y})^{n}=\frac{x^{n}}{y^{n}}$
- $x^{0}=1,\,x\neq 0$

IMPORTANCE

Understanding the laws of indices plays an important role in manipulating and simplifying expressions involving indices.
Laws of indices can also be used to evaluate expressions involving indices without the use of calculators.

DISCUSSION

BASE AND INDEX OF A NUMBER

It is vital to get familiarized with the base and index of a number. This will become the foundation of learning the laws of indices. In the given example, 3 is raised to 5. Meaning, the base will be multiplied to itself five times.

If we will evaluate the given, 3 x 3 x 3 x 3 x 3 = 243.

DISCUSSION

LAWS OF INDICES

Multiplication Law/ Product Rule – When expressions with the same base (x) are multiplied, the indices (m and n) are added.

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

Examples:

1) $2^{a}\cdot 2^{b}=2^{a+b}$ 2) $4^{\frac{1}{2}}\cdot 4^{\frac{1}{3}}=4^{\frac{1}{6}}$ 3) $z^{5}\cdot z^{4}=z^{9}$

Division Law/ Quotient Rule – When expressions with the same base (x) are divided, the indices (m and n) are subtracted.

$x^{m}/x^{n}=\frac{x^{m}}{x^{n}}=x^{m-n}$

Examples:

1) $5^{3a}/5^{2b}=5^{3a-2b}$ 2) $10^{6}/10^{2}=10^{6-2}=10^{4}$ 3) $y^{2a}/y^{4}=y^{2a-4}$

Power Law/ Power Rule – When an expression is raised to a certain index and is raised again to another index, the indices (m and n) are multiplied.

$(x^{m})^{n}=x^{m\cdot n}$

Examples:

1) $(6^{5p})^{2q}=6^{5p\cdot 2q}$ 2) $(9^{8})^{2}=9^{8\cdot 2}=9^{16}$ 3) $(c^{4})^{n}=c^{4\cdot n}=c^{4n}$

ILLUSTRATIVE EXAMPLES

Find the value of the following series.

$(c^{2}\cdot c^{8})/c^{5}=c^{10}/c^{5}=c^{10-5}=c^{5}$
$\frac{6^{13}}{6^{2}\cdot 6^{3}}=\frac{6^{13}}{6^{2(3)}}=\frac{6^{13}}{6^{6}}=6^{13-6}=6^{7}$
$m^{-2}n^{3}\cdot m^{5}n^{-4}=m^{-2+5}n^{3-4}=m^{3}n^{-1}=\frac{m^{3}}{n}$
$(p^{3}q^{5})^{3}=(p^{3(3)}q^{5(3)})=p^{9}q^{15}$
$3(4x^{4}y^{3})^{2}=3(4x^{4(2)}y^{3(2)})=3(4x^{8}y^{6})=12x^{8}3y^{6}$
$\frac{rs^{2}t^{3}}{s^{-5}t^{2}}=rs^{2+5}t^{3-2}=rs^{7}t$
$\sqrt[7]{\frac{b^{4}}{c^{-2}}}=b^{\frac{4}{7}}c^{\frac{2}{7}}$
$\frac{a^{2}b^{\frac{1}{3}}}{a^{\frac{1}{2}}b^{3}}=a^{2-\frac{1}{2}}b^{\frac{1}{3}-3}=\frac{a\frac{3}{2}}{b^{\frac{8}{3}}}$

REFERENCES: