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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Multiplying Monomials

After studying this lesson, you will be able to:

  • Multiply monomials.
  • Multiply powers with the same base.
  • Raise a power to a power.

Monomials have one term. The term can be a number, a variable, or the product of a number and a variable. Monomials are expressions with do not contain a + or - sign.... it has only one term.

 

Multiplying Powers with the Same Base:

The base stays the same; Add exponents

We are now ready for the next exponent rule:

Power of a Power: Raise the coefficient to the power; Multiply the exponents The power of a power rule is used when we are raising one power to another power.

 

Example 1

( x 2 ) 3

In this problem we have x 2 raised to the 3 rd power. Our coefficient is 1. Since we have parentheses, everything in the parentheses must be raised to the power of 3. If we raise the coefficient (1) to the 3 rd power, we have 1. Using the power of a power rule, we multiply the exponents 2 times 3.

Therefore, our answer is 1x 6 or x 6

 

Example 2

( x y 2 z 3 ) 2

In this problem we have x y 2 z 3 raised to the 2 nd power. Our coefficient is 1. Since we have parentheses, everything in the parentheses must be raised to the power of 2. If we raise the coefficient (1) to the 2 nd power, we have 1. Using the power of a power rule, we multiply the all the exponents in the parentheses by 3.

Therefore, our answer is x 2 y 4 z 6

 

Example 3

( 2x ) 3

In this problem we have 2x raised to the 3 rd power. This time our coefficient is 2. Since we have parentheses, everything in the parentheses must be raised to the power of 3. If we raise the coefficient (2) to the 3 rd power, we have 8 because 2 times 2 times 2 is eight. Using the power of a power rule, we multiply the exponents 1 times 3.

Therefore, our answer is 8x 3

 

Example 4

( -4 x y 2 ) 2

In this problem we have -4 x y 2 raised to the 2 nd power. Our coefficient is -4. Since we have parentheses, everything in the parentheses must be raised to the power of 2. If we raise the coefficient (-4) to the 2 nd power, we have 16 because -4 times -4 is sixteen. Using the power of a power rule, we multiply the exponents 1 and 2 by 2.

Therefore, our answer is 16 x 2 y 4

 

For the next group of problems, we will combine what we've learned in this section with the order of operations. Remember to use the correct order of operations:

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

 

Example 5

( 2x 2 ) 3 ( 3x 4 )

In this problem we have 2x 2 being raised to the3 rd power. Then we will need to multiply that answer by 3x 4 . We do the first parentheses first because it is being raised to a power. Following the order of operations, we always do the exponents before we multiply. 2x 2 being raised to the 3 rd power will give us 8x 6.

Next, we will multiply 8x 6 · 3x 4

Remember the rules for multiplying. We multiply the coefficients (8 and 3) and we add the exponents (6 and 4).

This will give us the answer: 24x 10

 

Example 6

( 6ab 2 ) 3 ( 5a ) 2

In this problem we have 6ab 2 being raised to the 3 rd power and we have 5a being raised to the 2 nd power. We do the first parentheses first because both are being raised to a power. Following the order of operations, we always do the exponents before we multiply. 6ab 2 being raised to the 3 rd power will give us 216 a 3 b 6 . 5a being raised to the 2 nd power will give us 25 a 2

Next, we will multiply 216 a 3 b 6 · 25 a 2 to giveus 5400 a 5 b 6 Remember the rules for multiplying. We multiply the coefficients (216 and 25) and we add the exponents.