## Power equations

### Step 1 :

1.1Negative number raised to an odd power is negative

To show this we begin with (-7)^{5} ,a negative number raised to an odd exponent:

(-7)^{5} can be written as (-7)•(-7)•(-7)•(-7)•(-7)

Now, using the rule that says minus times minus is plus, (-7)^{5} can be written as (49)•(49)•(-7) which is a negative number.

We showed that (-7)^{5} is a negative number

Using the same arguments as above, replacing (-7) by any negative number, and replacing the exponent 5 by any odd exponent, we proved which had to be proved

1.2 32 = 2^{5}(-32)^{1} = (2^{5})^{1} = -2^{5}

#### Equation at the end of step 1 :

(16^{1}) -2

^{5}————— - ——— 4 5

### Step 2 :

-2^{5}Simplify ——— 5

#### Equation at the end of step 2 :

(16^{1}) -32 ————— - ——— 4 5

### Step 3 :

3.1 16 = 2^{4}

(16)^{1} = (2^{4})^{1} = 2^{4}

#### Equation at the end of step 3 :

2^{4}-32 —— - ——— 4 5

### Step 4 :

2^{4}Simplify —— 4

#### Dividing exponents :

4.1 2^{4} divided by 2^{2} = 2^{(4 - 2)} = 2^{2}

#### Equation at the end of step 4 :

-32 4 - ——— 5### Step 5 :

#### Rewriting the whole as an Equivalent Fraction :

5.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using 5 as the denominator :

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

#### Adding fractions that have a common denominator :

5.2 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

### Final result :

52 —— = 10.40000 5## Simplifying Fractions Calculator

### Calculator Use

Convert improper fractions to mixed numbers in simplest form. This calculator also simplifies proper fractions by reducing to lowest terms and showing the work involved.

In order to simplify a fraction there must be:

- A number that will divide evenly into both the numerator and denominator so it can be reduced, or
- The numerator must be greater than the denominator, (an improper fraction), so it can be converted to a mixed number.

### What is an Improper Fraction?

An improper fraction is any fraction where the numerator is greater than the denominator. Examples of improper fractions are 16/3, 81/9, 525/71.

### How to Convert an Improper Fraction to a Mixed Number

- Divide the numerator by the denominator
- Write down the whole number result
- Use the remainder as the new numerator over the denominator. This is the fraction part of the mixed number.

Example: Convert the improper fraction 16/3 to a mixed number.

- Divide 16 by 3: 16 ÷ 3 = 5 with remainder of 1
- The whole number result is 5
- The remainder is 1. With 1 as the numerator and 3 as the denominator, the fraction part of the mixed number is 1/3.
- The mixed number is 5 1/3. So 16/3 = 5 1/3.

When possible this calculator first reduces an improper fraction to lowest terms before finding the mixed number form.

Example: Convert the improper fraction 45/10 to a mixed number.

- This calculator reduces the improper fraction to lowest terms by dividing numerator and denominator by the greatest common factor (GCF). The GCF of 45 and 10 is 5.
\( \dfrac{45\div5}{10\div5} = \dfrac{9}{2}\)

- Use this reduced improper fraction and divide 9 by 2: 9 ÷ 2 = 4 with remainder of 2
- The whole number result is 4
- The remainder is 1. With 1 as the numerator and 2 as the reduced denominator, the fraction part of the mixed number is 1/2.
- The mixed number 4 1/2. So 45/10 = 4 1/2.

### Related Calculators

For additional explanation of factoring numbers to find the greatest common factor (GCF) see the Greatest Common Factor Calculator.

If your improper fraction numbers are large you can use the Long Division with Remainders Calculator to find whole number and remainder values when simplifying fractions by hand.

To perform math operations on fractions before you simplify them try our Fractions Calculator. This calculator will also simplify improper fractions into mixed numbers.

16/1 is already in the simplest form. It can be written as 16 in decimal form (rounded to 6 decimal places).

#### Steps to simplifying fractions

- Find the GCD (or HCF) of numerator and denominator

GCD of 16 and 1 is**1** - Divide both the numerator and denominator by the GCD

16 ÷ 1/1 ÷ 1 - Reduced fraction: 16/1

Therefore, 16/1 simplified to lowest terms is 16/1.

MathStep (Works offline)

Download our mobile app and learn to work with fractions in your own time:

Android and iPhone/ iPad

Equivalent fractions: ^{32}/_{2}^{48}/_{3}^{80}/_{5}^{112}/_{7}

More fractions: ^{32}/_{1}^{16}/_{2}^{48}/_{1}^{16}/_{3}^{17}/_{1}^{16}/_{2}^{15}/_{1}

#### Fractions Simplifier

Sours: https://answers.everydaycalculation.com/simplify-fraction/16-1## Adding, subtracting and finding the least common multiple

### Step 1 :

1 Simplify — x#### Equation at the end of step 1 :

1 1 1 1 (——-————) ÷ (—+—) 16 (x^{2}) 4 x

### Step 2 :

1 Simplify — 4#### Equation at the end of step 2 :

1 1 1 1 (——-————) ÷ (—+—) 16 (x^{2}) 4 x

### Step 3 :

#### Calculating the Least Common Multiple :

3.1 Find the Least Common Multiple

The left denominator is : 4

The right denominator is : x

Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
---|---|---|---|

2 | 2 | 0 | 2 |

Product of all Prime Factors | 4 | 1 | 4 |

Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
---|---|---|---|

x | 0 | 1 | 1 |

Least Common Multiple:

4x

#### Calculating Multipliers :

3.2 Calculate multipliers for the two fractions

Denote the Least Common Multiple by L.C.M

Denote the Left Multiplier by Left_M

Denote the Right Multiplier by Right_M

Denote the Left Deniminator by L_Deno

Denote the Right Multiplier by R_Deno

Left_M = L.C.M / L_Deno = x

Right_M = L.C.M / R_Deno = 4

#### Making Equivalent Fractions :

3.3 Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example : 1/2 and 2/4 are equivalent, y/(y+1)^{2} and (y^{2}+y)/(y+1)^{3} are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

#### Adding fractions that have a common denominator :

3.4 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

#### Equation at the end of step 3 :

1 1 (x + 4) (—— - ————) ÷ ——————— 16 (x^{2}) 4x

### Step 4 :

1 Simplify —— x^{2}

#### Equation at the end of step 4 :

1 1 (x + 4) (—— - ——) ÷ ——————— 16 x^{2}4x

### Step 5 :

1 Simplify —— 16#### Equation at the end of step 5 :

1 1 (x + 4) (—— - ——) ÷ ——————— 16 x^{2}4x

### Step 6 :

#### Calculating the Least Common Multiple :

6.1 Find the Least Common Multiple

The left denominator is : 16

The right denominator is : x^{2}

Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
---|---|---|---|

2 | 4 | 0 | 4 |

Product of all Prime Factors | 16 | 1 | 16 |

Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
---|---|---|---|

x | 0 | 2 | 2 |

Least Common Multiple:

16x^{2}

#### Calculating Multipliers :

6.2 Calculate multipliers for the two fractions

Denote the Least Common Multiple by L.C.M

Denote the Left Multiplier by Left_M

Denote the Right Multiplier by Right_M

Denote the Left Deniminator by L_Deno

Denote the Right Multiplier by R_Deno

Left_M = L.C.M / L_Deno = x^{2}

Right_M = L.C.M / R_Deno = 16

#### Making Equivalent Fractions :

6.3 Rewrite the two fractions into equivalent fractions

^{2}—————————————————— = ———— L.C.M 16x

^{2}R. Mult. • R. Num. 16 —————————————————— = ———— L.C.M 16x

^{2}

#### Adding fractions that have a common denominator :

6.4 Adding up the two equivalent fractions

^{2}- (16) x

^{2}- 16 ————————— = ——————— 16x

^{2}16x

^{2}

#### Equation at the end of step 6 :

(x^{2}- 16) (x + 4) ————————— ÷ ——————— 16x

^{2}4x

### Step 7 :

x^{2}-16 x+4 Divide ————— by ——— 16x

^{2}4x

7.1 Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

^{2}- 16 x + 4 x

^{2}- 16 4x ——————— ÷ ————— = ——————— • ——————— 16x

^{2}4x 16x

^{2}(x + 4)

#### Trying to factor as a Difference of Squares :

7.2 Factoring: x^{2} - 16

Theory : A difference of two perfect squares, A^{2} - B^{2} can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A^{2} - AB + BA - B^{2} =

A^{2}- AB + AB - B^{2} =

A^{2} - B^{2}

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 16 is the square of 4

Check : x^{2} is the square of x^{1}

Factorization is : (x + 4) • (x - 4)

#### Canceling Out :

7.3 Cancel out (x + 4) which appears on both sides of the fraction line.

#### Dividing exponential expressions :

7.4 x^{1} divided by x^{2} = x^{(1 - 2)} = x^{(-1)} = ^{1}/_{x1} = ^{1}/_{x}

### Final result :

x - 4 ————— 4x## 1/2 simplify 16

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