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2

## How do you express 56,700,000 in scientific notation?

Shwetank Mauria
Featured 3 weeks ago

$56 , 700 , 000 = 5.67 \times {10}^{7}$

#### Explanation:

In scientific notation, we write a number so that it has single digit to the left of decimal sign and is multiplied by an integer power of $10$.

Note that moving decimal $p$ digits to right is equivalent to multiplying by ${10}^{p}$ and moving decimal $q$ digits to left is equivalent to dividing by ${10}^{q}$.

Hence, we should either divide the number by ${10}^{p}$ i.e. multiply by ${10}^{- p}$ (if moving decimal to right) or multiply the number by ${10}^{q}$ (if moving decimal to left).

In other words, it is written as $a \times {10}^{n}$, where $1 \le a < 10$ and $n$ is an integer.

To write $56 , 700 , 000$ in scientific notation, we will have to move the decimal point seven points to the left, which literally means dividing by ${10}^{7}$.

Hence in scientific notation $56 , 700 , 000 = 5.67 \times {10}^{7}$ (note that as we have moved decimal seven points to the left we are multiplying by ${10}^{7}$.

1

## How do you evaluate -( \frac { 1} { 2} ) ^ { 2} + 3+ 2?

Gimpy C.
Featured 3 weeks ago

$\frac{19}{4}$

#### Explanation:

$- {\left(\frac{1}{2}\right)}^{2} + 3 + 2$

We want to follow order of operations. The first thing to check is parentheses. There's a (1/2) in there, but that's as far as that can go. So, we check the next thing, which is exponents.

We must square both the numerator and denominator.

$- {1}^{2} / {2}^{2} + 3 + 2$

$- \frac{1}{4} + 3 + 2$

We are adding everything now, and that operation is communitive (we can do it in any order), but we have a fraction. Let's take care of the simple addition first.

$- \frac{1}{4} + 5$

With that out of the way, we need to make the 5 compatible by multiplying it by a factor of 1. Since our fraction is in fourths, we will multiply by $\frac{4}{4}$

$- \frac{1}{4} + 5 \cdot \frac{4}{4}$

$- \frac{1}{4} + \frac{20}{4}$

This can be written as

$\frac{20}{4} - \frac{1}{4}$

$\frac{19}{4}$

19 is prime, so we cannot do anything else. If we are not allowed to leave it as an improper fraction (larger numerator than denominator), we must convert to a mixed fraction.

To do this, we divide the numerator by the denominator to get the whole number, and the remainder is under the denominator unchanged.

$\frac{19}{4} = 4 r 3$

$4 \frac{3}{4}$

2

## What is 5/8 +7/8 ?

Jim G.
Featured 3 weeks ago

$\frac{12}{8} = \frac{3}{2}$

#### Explanation:

To add 2 fractions we require the $\textcolor{b l u e}{\text{denominators}}$ to be the same value.

In this case they are, both 8

We can therefore $\textcolor{b l u e}{\text{add the numerators}}$ while leaving the denominator as it is.

$\Rightarrow \frac{5}{8} + \frac{7}{8}$

$= \frac{5 + 7}{8}$

$= \frac{12}{8}$

We can $\textcolor{b l u e}{\text{simplify}}$ the fraction by dividing the numerator/denominator by the $\textcolor{b l u e}{\text{highest common factor}}$ of 12 and 8, which is 4

rArr12/8=(12÷4)/(8÷4)=3/2larrcolor(red)" in simplest form"

This process is normally done using $\textcolor{b l u e}{\text{cancelling}}$

$\Rightarrow \frac{12}{8} = {\cancel{12}}^{3} / {\cancel{8}}^{2} = \frac{3}{2} \leftarrow \textcolor{red}{\text{ in simplest form}}$

A fraction is in $\textcolor{b l u e}{\text{simplest form}}$ when no other factor but 1 divides into the numerator/denominator.

2

## Explain what is happening when using the difference method for determining the greatest common factor. Why does this work?

Tony B
Featured 3 weeks ago

See the explanation

#### Explanation:

$\textcolor{b l u e}{\text{The numeric reference}}$

Let one of the common factors be $f = 8$
let a numeric count be $n$

As the numbers to be tested I chose:

$8 \times 20 = 160$
$8 \times 15 = 120$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{The underlying principle}}$

As the process is based on subtraction then the starting point of

$160 - 120$ has to have a difference that is related to one of the factors. In that: $\text{ "120+nxx"some factor of 160} = 160$

This will be true of every subtraction in that the difference will a factor.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{The demonstration of process}}$

Set the following
$8 \times 20 = 160 = 20 f$
$8 \times 15 = 120 = 15 f$

The subtraction process

$20 f - 15 f = \textcolor{w h i t e}{1} 5 f \leftarrow \text{ largest - smallest: next use the 15 & 5}$

$15 f - \textcolor{w h i t e}{1} 5 f = 10 f \leftarrow \text{ largest - smallest: next use the 10 & 5}$

$10 f - \textcolor{w h i t e}{1} 5 f = \textcolor{w h i t e}{1} 5 f \leftarrow \text{ largest - smallest: next use the 5 & 5}$

$5 f - 5 f = 0 \leftarrow \text{ we have to stop at this point}$

This system is stating that the $G C F = 5 f = 5 \times 8 = 40$
.......................................................................................................
$\textcolor{b r o w n}{\text{Numeric equivalent}}$

$160 - 120 = 40 \text{ .......} \to \textcolor{w h i t e}{.} 5 f \to \textcolor{w h i t e}{.} 5 \times 8 = 40$
$120 - 40 = 80 \text{ .........} \to 10 f \to 10 \times 8 = 80$
$80 - 40 = 40 \text{ ...........} \to \textcolor{w h i t e}{.} 5 f \to \textcolor{w h i t e}{.} 5 \times 8 = 40$
$40 - 40 = 0$

$G C F = 40$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Using prime factor trees}}$

$G C F = 2 \times 2 \times 2 \times 5 = 40$

1

## How do you evaluate \frac { 27^ { \frac { 2} { 3} } } { 27^ { \frac { 1} { 3} } }?

Tony B
Featured 2 weeks ago

3

#### Explanation:

$\frac{1}{{27}^{\frac{1}{3}}}$ is the same as ${27}^{- \frac{1}{3}}$

So we can write: ${27}^{\frac{2}{3}} \times {27}^{- \frac{1}{3}}$

Another way of writing this is:

${27}^{\frac{2}{3} - \frac{1}{3}} \text{ "=" } {27}^{\frac{1}{3}}$

Which is the same as $\sqrt[3]{27}$

but $3 \times 3 \times 3 = 27$

so ${27}^{\frac{1}{3}} = \sqrt[3]{27} = 3$

1

## How do you evaluate \frac { 7} { 3} \cdot \frac { 15} { 14}?

Jim G.
Featured 2 weeks ago

$\frac{5}{2}$

#### Explanation:

There are 2 possible approaches to evaluating this product.

• color(red)" Simplify then multiply"larr" preferable method"

To simplify consider $\textcolor{b l u e}{\text{common factors}}$ of the values on the numerators with values on the denominators and $\textcolor{b l u e}{\text{cancel}}$

In this case 7 and 14 can be divided by 7 and 3 and 15 by 3

$\Rightarrow {\cancel{\textcolor{red}{7}}}^{1} / {\cancel{\textcolor{m a \ge n t a}{3}}}^{1} \times {\cancel{\textcolor{m a \ge n t a}{15}}}^{5} / {\cancel{\textcolor{red}{14}}}^{2} \leftarrow \text{ cancelling}$

$= \frac{1 \times 5}{1 \times 2}$

$= \frac{5}{2} \leftarrow \textcolor{p u r p \le}{\text{ in simplest form}}$

• color(red)" Multiply then simplify"

$\frac{7}{3} \times \frac{15}{14} = \frac{7 \times 15}{3 \times 14} = \frac{105}{42}$

If you see that 21 is the $\textcolor{b l u e}{\text{highest common factor}}$ then straight to the simplification.

$\frac{105}{42} = {\cancel{105}}^{5} / {\cancel{42}}^{2} = \frac{5}{2}$

If not then simplify in steps using 3 then 7, for example.

$\Rightarrow {\cancel{105}}^{35} / {\cancel{42}}^{14} = {\cancel{35}}^{5} / {\cancel{14}}^{2} = \frac{5}{2} \leftarrow \textcolor{p u r p \le}{\text{ in simplest form}}$

A fraction is in $\textcolor{p u r p \le}{\text{simplest form}}$ when no other factor but 1 will divide into the numerator/denominator.

1

## How do you evaluate 9( - 2) ^ { 2} - 5( 6) - 11?

Jim G.
Featured 1 week ago

$- 5$

#### Explanation:

2 points of note.

• 9(-2)^2=9xx(-2)^2

• -5(6)=-5xx6

When evaluating expressions with $\textcolor{b l u e}{\text{mixed operations}}$ there is a particular order that must be followed.

Follow the order as set out in the acronym PEMDAS

[Parenthesis (brackets), Exponents (powers), Multiplication, Division, Addition, Subtraction ]

$\Rightarrow 9 {\left(- 2\right)}^{2} - 5 \left(6\right) - 11$

$= \left(9 \times 4\right) - 5 \left(6\right) - 11 \leftarrow \textcolor{red}{\text{Exponents}}$

$= 36 - 30 - 11 \leftarrow \textcolor{red}{\text{Multiplication}}$

Subtract from left to right.

$= - 5 \leftarrow \textcolor{red}{\text{Subtraction}}$

1

## How do you subtract 5 1/16 - 1 13/18?

Jim G.
Featured 1 week ago

$3 \frac{49}{144}$

#### Explanation:

There are 2 possible approaches to this calculation, both made fairly 'awkward' due to the values on the denominators of the fractions.

$\textcolor{red}{\text{Approach 1}}$

Change the $\textcolor{b l u e}{\text{mixed numbers " "to "color(blue)"improper fractions}}$

$\Rightarrow 5 \frac{1}{16} = \frac{81}{16} \text{ and } 1 \frac{13}{18} = \frac{31}{18}$

The calculation is now.

$\frac{81}{16} - \frac{31}{18}$

Before we can subtract the fractions we require them to have a
$\textcolor{b l u e}{\text{common denominator}}$

We have to find the $\textcolor{b l u e}{\text{lowest common multiple}}$ ( LCM) of 16 and 18

The LCM of 16 and 18 is 144

$\Rightarrow \frac{81}{16} \times \frac{9}{9} = \frac{729}{144} \text{ and } \frac{31}{18} \times \frac{8}{8} = \frac{248}{144}$

$\Rightarrow \frac{729}{144} - \frac{248}{144} \leftarrow \textcolor{red}{\text{ is now the calculation}}$

Since the denominators are now common we can subtract the numerators, leaving the denominator as it is.

$\Rightarrow \frac{729}{144} - \frac{248}{144} = \frac{729 - 248}{144}$

$= \frac{481}{144} = 3 \frac{49}{144} \leftarrow \textcolor{red}{\text{ returning a mixed number}}$

$\textcolor{red}{\text{Approach 2}}$

"Using the fact that "5 1/16=5+1/16;1 13/18=1+13/18

$\text{Then } 5 \frac{1}{16} - 1 \frac{13}{18}$

$= 5 + \frac{1}{16} - \left(1 + \frac{13}{18}\right) = 5 + \frac{1}{16} - 1 - \frac{13}{18}$

We can now subtract the numbers and subtract the fractions separately.

$\Rightarrow 5 + \frac{1}{16} - 1 - \frac{13}{18} = \left(5 - 1\right) + \frac{1}{16} - \frac{13}{18}$

$= 4 + \left(\frac{1}{16} \times \frac{9}{9} - \frac{13}{18} \times \frac{8}{8}\right)$

$= 4 + \left(\frac{9}{144} - \frac{104}{144}\right)$

$= 4 + \left(- \frac{95}{144}\right)$

$= 4 - \frac{95}{144}$

$= \frac{576}{144} - \frac{95}{144}$

$= \frac{481}{144}$

$= 3 \frac{49}{144} \leftarrow \textcolor{red}{\text{ as a mixed number}}$

1

## What is the reciprocal of 5?

EZ as pi
Featured 1 week ago

The reciprocal is $\frac{1}{5}$

#### Explanation:

The reciprocal of a number is also called its multiplicative inverse.

To find the reciprocal you flip the number.

$5 = \frac{5}{1}$ as a fraction

The reciprocal is $\frac{1}{5}$

When you multiply a number by its reciprocal the answer is always $1$ which is the identity element for multiplication and division.

$\frac{5}{1} \times \frac{1}{5} = 1$

$\frac{a}{1} \times \frac{1}{a} = 1 \text{ }$ assuming that $a \ne 0$

1

## How do you evaluate \frac { 1} { 6+ \frac { 1} { 8} }?

Barney V.
Featured 4 days ago

$\frac{8}{49}$

#### Explanation:

$\frac{1}{6 + \frac{1}{8}}$

$\therefore = \frac{1}{6 \frac{1}{8}}$

$\therefore = \frac{1}{\frac{49}{8}}$

$\therefore = \frac{1}{1} \times \frac{8}{49}$

$\therefore = \frac{8}{49}$

check:

$\frac{1}{8} = 0.125 + 6 = 6.125$

$1 \div 6.125 = 0.163265306$

$\frac{8}{49} = 0.163265306$