exponent

Examples of exponent in the following topics:

• Negative Exponents

  • Solving mathematical problems involving negative exponents may seem daunting.
  • However, negative exponents are treated much like positive exponents when applying the rules for operations.
  • Note that if we apply the rule for division of numbers with exponents, we have:
  • This rule makes it possible to simplify expressions with negative exponents.
  • Note that each of the rules for operations on numbers with exponents still apply when the exponent is a negative number.

• Rational Exponents

  • Rational exponents are another method for writing radicals and can be used to simplify expressions involving both exponents and roots.
  • In such cases, the exponent acts as both a whole number exponent and a root, or fraction exponent.
  • We can simplify the fraction in the exponent to 2, giving us $5^2=25$.
  • Recall the rule for dividing numbers with exponents, in which the exponents are subtracted.
  • Relate rational exponents to radicals and the rules for manipulating them

• Simplifying Exponential Expressions

  • The rules for operating on numbers with exponents can be applied to variables with exponents as well.
  • Recall the rules for operating on numbers with exponents, which are used when simplifying and solving problems in mathematics.
  • For example, consider the rule for multiplying two numbers with exponents.
  • To simplify the second part of the expression, apply the rule for multiplying numbers with exponents:
  • We can also apply the rule for raising a power to another exponent:

• Introduction to Exponents

  • Here, the exponent is 3, and the expression can be read in any of the following ways:
  • Some exponents have their own unique pronunciations.
  • Let's look at an exponential expression with 2 as the base and 3 as the exponent:
  • Any number raised by the exponent $1$ is the number itself.
  • Any nonzero number raised by the exponent 0 is 1. 

• Simplifying Algebraic Expressions

  • For example, look at this figure, as you can see, the expression consists of an exponent, coefficients, terms, operators, constants and variables.
  • Usually terms with the highest power (exponent), are written on the left.
  • Likewise when the exponent (power) is one.
  • When the exponent is zero, the result is always 1.
  • Multiplied terms are simplified using exponents.

• Logarithmic Functions

  • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
  • In its simplest form, a logarithm is an exponent.
  • Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more.
  • Here we are looking for the exponent to which $3$ is raised to yield $243$.
  • More recently, logarithms are most commonly used to simplify complex calculations that involve high-level exponents.

• Scientific Notation

  • Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved.
  • Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.
  • The exponent is -4 because the decimal point was moved to the left (the exponent would be positive had the decimal been moved to the right) by exactly 4 places.A number written in scientific notation can also be converted to standard form by reversing the process described above.
  • This form allows easy comparison of two numbers of the same sign with $m$ as a base, as the exponent $n$ gives the number's order of magnitude.
  • Because superscripted exponents like $10^7$ cannot always be conveniently displayed, the letter E or e is often used to represent the phrase "times ten raised to the power of" (which would be written as "$\cdot 10^n$") and is followed by the value of the exponent.

• Adding, Subtracting, and Multiplying Radical Expressions

  • Radicals and exponents have particular requirements for addition and subtraction while multiplication is carried out more freely.
  • Roots are the inverse operation for exponents.
  • It's easy, although perhaps tedious, to compute exponents given a root.
  • Let's go through some basic mathematical operations with radicals and exponents.
  • the value under the radical sign can be written as an exponent,

• Rules for Exponent Arithmetic

  • There are rules for operating on numbers with exponents that make it easy to simplify and solve problems.
  • There are several useful rules for operating on numbers with exponents.
  • Note that you can only add exponents in this way if the corresponding terms have the same base.
  • For the first part of the expression, apply the rule for a product raised to an exponent:
  • Explain and implement the rules for operating on numbers with exponents

• Converting between Exponential and Logarithmic Equations

  • Here we are looking for the exponent to which $3$ is raised to yield $243$.
  • As the exponent and log on the left side of the equation undo each other we are left with:
  • An exponential equation is an equation where the variable we are solving for appears in the exponent.
  • Here since the bases are both $5$, the exponents are equal.
  • Next we use the properties of logarithms to move the variable out of the exponent.