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Explain why (–4x)^0 = 1, but –4x^0 = –4.

I. (–4x)^0 = 1

$$1=\dfrac{(-4x)^1}{(-4x)^1}=(-4x)^1\cdot (-4x)^{-1}=(-4x)^{1-1}=(-4x)^0$$

II. –4x^0 = –4

$$-4 = -4\cdot 1 = -4 \cdot \dfrac { x^1 } { x^1 } = -4 \cdot x^1 \cdot x^{-1} = -4 \cdot x^{1-1} = -4 \cdot x^0$$

This is due to a readability issue. You see, normally you follow an order of significance: first you do powers, then you do multiplications, then additions (and the inverse of each happens at the same time, and goes from left to right).

In the first case, you got (-4 * x)^0

Well anything to the power of 0 = 1 so clearly this equation is: (-4 * x)^0 = 1

In the second case, you got -4 * x ^0

The way this must be read is doing the powers first, so we get -4 * 1

This then multiplies and gives us: -4 * x ^0 = -4

Remember: it never hurts to use more parenthesies if you really want to make it clear what order you meant. It's important to make sure the equation is easy to read.

what is the 20th term if the first term is 4 and the common difference is -1/3

$$\small{
\text{d} = -\dfrac13 \qquad a_1 = 4
}\\\\
\small{
\text{arithmetical sequence: } a_n = a_1 + (n-1)\cdot d
}\\\\
\begin{array}{rclclcr}
\small{a_{20} &=& 4 + (20-1)\cdot \left( -\dfrac13 \right)\\\\
\small{a_{20} &=& 4 + 19\cdot \left(-\dfrac13 \right)\\\\
\small{a_{20} &=& 4 - \dfrac{19}{3} \\\\
\small{a_{20} &=& -2.\overline{3} \\\\
\end{array}$$

It appears the common ratio is -¹/₅

The sum of a geometric series is:

$${a}{\left({\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{{\mathtt{r}}}^{{\mathtt{n}}}\right)}{\left({\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{r}}\right)}}\right)}$$

Where:

a = first term

r = common ratio

n = Number of terms of sequence

So the sum of the first 7 terms is:

75 * (1 - (-¹/₅)⁷) / (1 - -¹/₅)

= 75 * (1 - -0.0000128) / (1 + ¹/₅)

= 75 * 1.0000128 / 1.2

= 62.5008

 find the sum of 7 terms of 75,-15,3,-3/5

$$\small{
\text{ratio} = -\dfrac15 = -0.2 \qquad a_1 = 75
}\\\\
\small{
\text{geometric sequence: } a_n = a_1 \cdot r^{n-1}
}\\\\
\begin{array}{rclclcr}
\small{a_1 &=& 75 \cdot r^0 &=& 75 }\\ \\
\small{a_2 &=& 75 \cdot r^1 &=& 75 \cdot \left( -\dfrac15 \right)^1 &=& -15 }\\\\
\small{a_3 &=& 75 \cdot r^2 &=& 75 \cdot \left( -\dfrac15 \right)^2 &=& 3 }\\\\
\small{a_4 &=& 75 \cdot r^3 &=& 75 \cdot \left( -\dfrac15 \right)^3 &=& -0.6 }\\\\
\small{a_5 &=& 75 \cdot r^4 &=& 75 \cdot \left( -\dfrac15 \right)^4 &=& 0.12 }\\\\
\small{a_6 &=& 75 \cdot r^5 &=& 75 \cdot \left( -\dfrac15 \right)^5 &=& -0.024 }\\\\
\small{a_7 &=& 75 \cdot r^6 &=& 75 \cdot \left( -\dfrac15 \right)^6 &=& 0.0048 }\\\\
\end{array}\\\\
\small{
\mathrm{sum}_7 = 75-15+3-0.6+0.12-0.024+0.0048= 62.5008
}\\\\
\small{
\mathrm{sum}_7 = a_1 \cdot \left( \dfrac{ 1-r^7} {1-r} \right)
= 75\cdot \left( \dfrac{ 1-(-0.2)^7} {1-(-0.2)} \right)
=75\cdot \left( \dfrac{ 1+0.2^7} {1.2)} \right)
=62.5008
}$$

$$\boxed{~~
\small{
\mathrm{sum}_n = a_1 \cdot \left( \dfrac{ 1-r^n} {1-r} \right)
}
~~}$$

its 4 for the startvalue + 20*-1/3 so => 4+20*(-1/3)= 2.666666

This is undefined in mathematics. 

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There are a lot more 9's in the 359.9... than there are 0's in the 360.0..., so 359.9999...... is closer to 360 than 360.000...etc.

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For % click on  1/x  and hold a little bit longer

or click on  2nd  and then  on  % !

Use asin, which is the same thing (press 2nd then asin)

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Haven't you missed out a decimal place in each number?!!!

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final velocity = intial velocity + acceleration*time, so

final velocity = 5 + 10*3 = 35 m/s

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13 multiplied by 1/2 is 6.5.

13 divided by 1/2 is the same as 13 multiplied by the inverse of 1/2, namely 2; so 13 divided by 1/2 is 13*2 = 26.  In other words it takes 26 halves to make 13.

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On the calculator here you can enter log(number, base) 

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You need to be more specific.  Is it the shape/size/...?  What dimensions do you know? etc.

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Notice the pattern......

(4-3) +( 8-6) + (12-9) + (16-12)+..........+ (96-72) + (100-75)   =

1 + 2 + 3 + 4 + ........+ 24  + 25  =     [25 * 26] / 2 =  25 * 13 =   325

We can use a proportion to solve this

2 / 25   =    x / 60      cross-multiply

120 = 25x      divide by 25 on both sides

120/25 =  x  =  24/5    =     4.8 inches

13 divided by half is 6.5

a^2 = 74.1     take the square root of both sides

a   = √74.1   ≈ 8.61 rounded to two decimal places  {to be fair, "a" could have also been the negative of this, as well..} 

thank u 

100(1.08)^20 is equal to 466.09 

In the second-to-the-last line, shouldn't the times sign be a minus sign?

That is:  12x  =  ln(y + 1)  -  ln(y - 1)

                x  =  [ ln(y + 1)  -  ln(y - 1) ] / 12

An example of conjugates is:  a + bi  and  a - bi

Multiply them as you would any two binomials, using First, Outside, Inside, Last:

(a + bi)(a - bi)  =  a·a + a(-bi) + (bi)a - (bi)(bi)

                       =  a² - abi + abi - b²i²

Since i² = -1:    =  a² - (b²)(-1)

                       =  a² + b²

This gives you a formula:  (a + bi)(a - bi)  =  a² + b²

Example:  (3 + 4i)(3 - 4i)  =  3² + 4²  =  9 + 16  =  25

Example:  (3x + 4yi)(3x - 4yi)  =  (3x)² + (4y)²  =  9x² + 16y²