Exponents

Welcome to the *astonishingly clear introduction* to exponents, roots, and logarithm. Learn in this

• representation of exponents

• roots and logarithms are two inverses of exponents

• difference between common and natural logarithms

• arithmetics with exponents and logarithms

• Precedence -- PEMDAS or BODMAS (Exponent and Order)

• introduction to squares and square roots

• introduction to cubes and cube roots

All the above are given in a simple thought process.

Representation of Exponents

In this page, exponent is introduced. We have studied that repeated addition is named multiplication.

Similarly, repeated multiplication is named exponents.

• ${2}^{3}=2\times 2\times 2=8$

• The expression $2}^{3$ is, $2$ to the power of $3$ or $2$ power $3$.

root : An inverse of Exponent

This page introduces "root".

One of the inverses of exponent is root. Root is introduced with the following two.

• first principle -- Root of a number to a given power of root is the base of the exponent with the given power.

• Simplified Procedure -- Root of a number is found from prime-factorization of the numbers (if root evaluates to an integer).
This introduction "root is an inverse of exponent" is *astoundingly clear and makes it simple for students*.

Logarithm : An inverse of Exponent

This page introduces "logarithm".

One of the inverses of exponent is logarithm. Logarithm is introduced with the following two.

• first principles -- Logarithm of a number is the power in the equivalent exponent.

• Simplified Procedure -- Logarithm of a number is found from prime-factorization of the number (if log evaluates to an integer).
This introduction "logarithm is an inverse of exponent" is *astoundingly clear and makes it simple for students*.

Common and Natural Logarithms

This page explains

• common logarithm or logarithm of base $10$

• natural logarithm or logarithm of base $e$

The number $e$ is also introduced in a *thought-provoking and ingenious discourse*.

Arithmetics with Exponents

Arithmetics with exponents, without evaluating the exponent, is explained. For example $a}^{m}\times {a}^{n}={a}^{m+n$. The list of formulas are derived using the first principles of exponent.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

Arithmetics of Logarithms

Arithmetics with logarithms, without evaluating the logarithm, is explained. For example ${\mathrm{log}}_{a}{b}^{m}=m{\mathrm{log}}_{a}b$.

The list of formulas are derived using the first principles of logarithms.

These known results are given as a set of formulas. Students are advised to work them out quickly using the first principles. No need to memorize, and if the formulas are used repeatedly, over time, these can be recalled quickly.

Understanding exponents, roots, logarithms

This topic is little advanced for high school students. The various possibilities for roots and logarithms are discussed.

• root with negative power ($\sqrt[-2]{4}$)

• root with power 0 ($\sqrt[0]{4}$)

• logarithm with fraction base (${\mathrm{log}}_{\frac{1}{2}}8$)

• logarithm of negative value (${\mathrm{log}}_{2}(-4)$)

• logarithm with negative base (${\mathrm{log}}_{-2}(-8)$)

• logarithm with 0 base (${\mathrm{log}}_{0}2$)

• logarithm with 1 base (${\mathrm{log}}_{1}2$)

Note: Some of the above are not defined and has no meaning. Read through the lesson to understand the details.

Exponents: Simplification of Expressions

In this page, handling of exponents, roots, and logarithms in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.

Exponents : Precedence Order PEMA

Redefining the precedence order in arithmetics with PEMA or BOMA.

This topic is repeated to make sure that it is clear to the students. The details are updated to explain handling exponents, roots, and logarithms.

Representation of Squares and Square Roots

In this page, square of a number and square root of a number are revised in a simple thought process. The long division method to find the square root is explained.

Representation of Cubes and Cube Roots

In this page, cube of a number and cube root of a number are revised in a simple thought process.