Free Algebra
Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Powers and roots


Powers are used when we want to multiply a number by itself repeatedly.

1. Powers

When we wish to multiply a number by itself we use powers, or indices as they are also called. For example, the quantity 7× 7×7×7 is usually written as . The number 4 tells us the number of sevens to be multiplied together. In this example, the power, or index, is 4. The number 7 is called the base.


6 = 6×6 = 36. We say that "6 squared is 36", or "6 to the power 2 is 36".

2 = 2×2×2×2×2. We say that "2 to the power 5 is 32".

Your calculator will be pre-programmed to evaluate powers. Most calculators have a button marked , or simply ^. Ensure that you are using your calculator correctly by verifying that 3 = 177147.

2. Square roots

When 5 is squared we obtain 25. That is 5 = 25.

The reverse of this process is called finding a square root. The square root of 25 is 5. This is written as , or simply .

Note also that when -5 is squared we again obtain 25, that is (-5) = 25. This means that 25 has another square root, -5.

In general, a square root of a number is a number which when squared gives the original number. There are always two square roots of any positive number, one positive and one negative. However, negative numbers do not possess any square roots.

Most calculators have a square root button, probably marked . Check that you can use your calculator correctly by verifying that , to four decimal places. Your calculator will only give the positive square root but you should be aware that the second, negative square root is -8.8882.

An important result is that the square root of a product of two numbers is equal to the product of the square roots of the two numbers. For example

More generally,

However your attention is drawn to a common error which students make. It is not true that . Substitute some simple values for yourself to see that this cannot be right.


1. Without using a calculator write down the value of

Find the square of the following:

3. Show that the square of is 50.


1. 18, (and also -18).

2. a) 2, b) 12.

3. Cube roots and higher roots

The cube root of a number, is the number which when cubed gives the original number. For example, because 4 = 64

we know that the cube root of 64 is 4, written All numbers, both positive and negative, possess a single cube root.

Higher roots are defined in a similar way: because 2 = 32, the fifth root of 32 is 2, written


1. Without using a calculator find


1. a) 3, b) 5.


Expressions involving roots, for example and are also known as surds. Frequently, in engineering calculations it is quite acceptable to leave an answer in surd form rather than calculatingits decimal approximation with a calculator.

It is often possible to write surds in equivalent forms. For example, can be written as , that is


1. Write the following in their simplest surd form:

2. By multiplying numerator and denominator by , show that


All Right Reserved. Copyright 2005-2023