Multiply Mixed Numbers

  Here in this page we are going to discuss about multiplying mixed numbers.Mixed numbers can be written in form of improper fraction. The mixed numbers have summation of whole number and proper fraction. Proper fraction is nothing but the numerator of the fraction is less that that the denominator of the fraction. By multiplying the mixed numbers, we must follow certain procedure as follow.

How to Multiply Mixed Numbers

Let's see the procedure-

Step 1: Convert the given mixed numbers to improper fraction.

To make the mixed numbers to improper fraction, we multiply the whole number and denominator of the fraction and add the result to the numerator of the fraction.

Example: 2 ¼

Multiply 2 and 4 we get 8

Add the number 8 to the numerator of the fraction 1, we get 8 + 1 = 9.

Now get the improper fraction of the mixed number 2 ¼ = `(9)/(4)`

Step 2: Multiply the numerator and denominator of the improper fraction separately.

Example: consider the two improper fractions.

`(a)/(b)` * `(c)/(d)` , where, a > b and c > d

`(a * c)/(b * d)`  = `(p)/(q)`

Step 3: Now convert the improper fraction by mixed number as following procedure.

Example: `(8)/(5)`

5) 8 ( 1

       5

_________

        3

___________

Mixed number can be written as, quotient as the whole number, remainder is the numerator of the proper fraction and divisor is the denominator of the proper fraction.

`(8)/(5)`  = 1`(3)/(5)`

Examples:

Here are the examples on Multiplying Mixed Numbers

Example 1:

Multiplying the mixed numbers 3 ½ and 6 ¼.

Solution:

Step 1: Convert the mixed numbers 3 ½ to improper fraction, we get

3 ½ = `(7)/(2)`

Step 2: Convert the mixed numbers 6 ½ to improper fraction, we get

6 ¼ =  `(25)/(4)`

Step 3: Multiplying the improper fractions  `(7)/(2)` and `(25)/(4)`, we get

`(7)/(2)` * `(25)/(4)` =  `(175)/(8)`

Step 4: Convert the improper fraction  `(175)/(8)` to mixed number.

8) 175 (21

    16

___________

       1 5

          8

___________

           7

Mixed number  `(175)/(8)` =  `21(7)/(8)`

Answer:  `21(7)/(8)`

Example 2:

Multiplying the mixed numbers 4 ½ and 2 ¾.

Solution:

Step 1: Convert the mixed numbers 4 ½ to improper fraction, we get

4 ½ = `(9)/(2)`

Step 2: Convert the mixed numbers 2 ¾ to improper fraction, we get

2 ¾ = `(11)/(4)`

Step 3: Multiplying the improper fractions `(9)/(2)` and `(11)/(4)`, we get

`(9)/(2)` * `(11)/(4)` = `(99)/(8)`

Step 4: Convert the improper fraction `(99)/(8)` to mixed number.

8) 99 (12

    8

___________

     1 9

     1 6

___________

         3

Mixed number `(99)/(8)` =   `12(3)/(8)`

Answer:   `12(3)/(8)`

Exponents Negative Numbers

Introduction:

         The exponent number is shows how many times to use the base number in a multiplication. The exponent number is placed at upper right of the base number. Exponent -x in the expression a^-x. For example, -4 is the exponent in 2^-4= 1/2⁴ = 0.0625. The base and exponent numbers may be positive or negative. Negative exponents are a way of indicating reciprocals.

Rules of Exponents Negative Numbers

Definitions

1. aⁿ = a·a·a···a  (n times)

2. a⁰ = 1  (a ≠ 0)

3. a^-1 = 1/aⁿ (a  ≠ 0)

4. a^m/n = ⁿ√a^m or (ⁿ√a)_m (a ≥ 0, m ≥ 0, n > 0)

Combining

1. Multiplication: a^x a^y = a^{x + y}

2. Division: a^x / a^y = a^x-y (a ≠ 0)

3. Powers: (a^x)^y = a^xy

Distributing   (a ≥ 0, b ≥ 0)

1. (ab)^x = a^x b^x

2. (a/b)^x = a^x/b^x  (b ≠ 0)

Careful!!

1. (a + b)ⁿ ≠ aⁿ + bⁿ

2. (a – b)ⁿ ≠ aⁿ – bⁿ

Rule for Exponents Negative Numbers:

a^-n = 1/aⁿ

Examples: 5^-2 = 1/5² = 1/25 

(2/3)^-3 = (3/2)³ = 27/8

Exponents Negative Numbers – Examples

Negative exponents numbers solved problems

Example 1: Solve this expression 8^-2

Solution:

     Here, the exponent is -2 negative exponents. Usually in positive exponent the exponent number is shows how many times to use the base number in a multiplication. In negative exponent number also shows like this. But in the negative exponent numbers we have to find the reciprocal of the numbers.

8^-2 = 1/8² = 1 / (8 × 8) = 1/64 = 0.015625

8² = 8 × 8 = 64.

Reciprocal of the number is 1/64 = 0.015625

Example 2: Solve 4^-3

Solution:

4^-3 = 1/4³ = 1 / (4 × 4 × 4) = 1/64 = 0.015625

Example 3: Solve this expression 2²/2^-3

Solution:

2²/2^-3 = 2² × 2³ = 2⁵ = 2 × 2 × 2 ×2 × 2 = 32

Example 4: Solve this expression 2(3^-1)

Solution:

2(3^-1) = 2(1/3) = 2/3 = 0.67

Example 5: Simplify this equation and solve this equation (4x)^-3, x = 2.

Solution:

(4x)^-3 = 1/64x³

Put x = 2

1/64(2³) = 1/64(8) = 1/512 = 0.001953125

Example 6: Simplify this equation and solve this equation (x^-3/y^-4)^-3, x = 1, y = 1.

Solution:

(x^-3 / y^-4)^-3 = (x^-3)^-3/(y^-4)^-3 = (y^-4)³/(x^-3)³ = y^-12/x^-9 = x⁹/y¹²

Put x =1 and y = 1 in the equation to get

1⁹/1¹² = 1/1 =1

Exponents Negative Numbers – Practice

Solve these problems for practice on negative exponents.

Problem 1: Solve 2^-2 - Answer: 0.25

Problem 2: Solve 3^-2 - Answer: 0.11

Problem 3: Solve 3²/3^-3 - Answer: 243

Problem 4: Solve this expression 4(4^-1) - Answer: 1

Problem 5: Simplify this equation and solve this equation (2x)^-2, x = 3. - Answer: 36

Precalculus Calculator Online

Introduction :

Precalculus calculator online is one interesting topics in mathematics. Precalculus calculator online is used to solve different types of precalculus problems. Calculator is a web-based tool to solve the problems. Online is nothing but the one computer is connected with another computer through a network or a cable. Here we solve some precalculus calculator online problems.

Example Problems for Online Precalculas Calculator:

Example problems for online precalculas calculator are given below:

Example 1:

Solve the quadratic equation x2 + x – 42.

Solution:

Let f(x) = x2 + x – 42

Now, plug f(x) = 0

x2 - 6x +7x - 42 = 0

x(x - 6) + 7(x - 6) = 0

(x - 6)(x + 7) = 0

x = 6; x = -7

The roots are x = 6, x = -7.

Example 2:

Solve 12x – 4y + 20 = 0. Find the slope and y-intercept for the given straight line.

Solution:

12x – 4y + 20 = 0

– 4y = – 12x – 20

Dividing by -4,

y = 3x + 5 ? (1)

General form of a straight line is,

y = mx + b ? (2)

Where, m = slope of a line,

b = y intercept of a line,

Here, y = 3x + 5

Compare the equation (1) and (2), we get,

Slope of the line m = 3,

y-intercept of the line b = 5.


Additional Example problems for online precalculas calculator are given below:

Example 3:

Find the center and radius of the circle for the given standard equation x2 + 10x + y2 – 8y – 7 = 0

Solution:

Given: x2 + 10x + y2 – 8y – 7 = 0

Standard equation for circle with center (a, b) and radius r is,

(x - a)2 + (y - b)2 = r2

Completing the x terms and y terms on the square that gives

(x2 + 10x + 10) + (y2 - 8y + 8) – 7 - 10 - 8 = 0

(x2 + 10x +10) + (y2 - 8y + 8) = 7 + 10 + 8

(x + 10)2 + (y - 8)2 = 25,

Solution to the center of the circle is (10, -8), and the radius is 5.

Example 4:

Find the vertex of the parabola y = 5x2 – 30x + 9

Solution:

General form:

x-coordinate for the vertex of the parabola is x = -b/2a,

y-coordinate is find by substitute the value for x into f(x)

Given:  y = 5x2 – 30x + 9

We know that x = -b/2a,

Here a = 5, b = -30

So that,   X = -b/2a = -(-30)/(2*5) = 3

And then y = 5(32) – 30(3) + 9 = 45 – 90 + 9 = -36

Solution to the problem is x = 3 and y = -36.