hectictar

3.          \(\, \ \, \ \, (\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{4}-\sqrt[3]{6}+\sqrt[3]{9}) \\ =(\sqrt[3]3)(\sqrt[3]{4})-(\sqrt[3]3)(\sqrt[3]{6})+(\sqrt[3]3)(\sqrt[3]{9}) + (\sqrt[3]{2})(\sqrt[3]{4})-(\sqrt[3]{2})(\sqrt[3]{6})+(\sqrt[3]{2})(\sqrt[3]{9}) \\ =\sqrt[3]{12}-\sqrt[3]{18}+\sqrt[3]{27} + \sqrt[3]{8}-\sqrt[3]{12}+\sqrt[3]{18} \\ =\sqrt[3]{27} + \sqrt[3]{8} \\ =3+2\\ =5\)

5.          \(\, \ \, \ \, \sqrt{10000}-\sqrt{81}+\sqrt{81} \\ =\sqrt{10000} \\ =\sqrt{10^4} \\ =10^2 \\ =100\)

6.     \(\frac{8}{x}=x+\frac1{x} \\~\\ 8=x^2+1 \\~\\ 7=x^2 \\ x=\pm\sqrt{7} \\ \text{ The smallest value is -sqrt(7) .}\)

4a² - 10a + 6  =  0       First, let's divide through by  2  .

2a² - 5a + 3  =  0

We can use the Quadratic formula to solve this problem for  a  .

The Quadratic formula says, if you have an equation written in the form

Ax² + Bx + C  =  0   , then...   \(x = {-{\color{blue}B} \pm \sqrt{{\color{blue}B}^2-4{\color{red}A}{\color{green}C}} \over 2{\color{red}A}}\)

So, for our problem...

2a² + -5a + 3  =  0

x = a ,  A = 2 ,  B = -5 ,  C = 3  .    Substitute these values into the Quadratic formula.

\(a = {-{\color{blue}(-5)} \pm \sqrt{{\color{blue}(-5)}^2-4{\color{red}(2)}{\color{green}(3)}} \over 2{\color{red}(2)}} \\~\\ a = {5 \pm \sqrt{25-24} \over 4} \\~\\ a={5 \pm \sqrt{1} \over 4} \\~\\ a={5 \pm 1 \over 4} \\~\\ \begin{array}\ a=\frac{5+1}{4}\qquad \text{ or }\qquad &a&=\frac{5-1}{4} \\~\\ a=\frac{6}{4}&a&=\frac44 \\~\\ a=\frac32&a&=1 \\~\\ \end{array}\)

I heard about that set of books before.........an interesting read, for sure!!!

Here's a sample...page 7 from volume 4:

It looks like a real page-turner !!!

In a way...it reminds me of another book—one written by John Daniel Edward Torrance....

We have a right triangle where  one leg = 1.94 m  and the  hypotenuse = 2.85 m  .

And...we can find the measure of angle  a  using the cosine.

cos a  =  adjacent / hypotenuse

cos a  =  1.94 / 2.85

                                              Take the arccosine of both sides.

a  =  arccos( 1.94 / 2.85 )

                                              Put this into a calculator to get that.....

a  ≈  47.1°

Express     \(\frac8{x-4}+\frac5{x^2+2x-24}\)     as a single fraction.

    \( \frac8{x-4}+\frac5{x^2+2x-24} \)

                                                  Factor the denominator of the second fraction.

                                                  What two numbers add to  2  and multiply to  -24 ?→  +6  and  -4  .

=  \( \frac8{x-4}+\frac5{(x+6)(x-4)} \)

                                                  Multiply the first fraction by \( \frac{x+6}{x+6} \)  .

=  \( (\frac{x+6}{x+6})(\frac8{x-4})+\frac5{(x+6)(x-4)} \)

=  \( \frac{8x+48}{(x+6)(x-4)}+\frac5{(x+6)(x-4)} \)

                                                  Now there's a common denominator so we can add the fractions.

=  \( \frac{8x+48+5}{(x+6)(x-4)} \)

=  \( \frac{8x+53}{(x+6)(x-4)} \)

The sum of the angles in a quadrilateral  =  360°     So.....

x   +   5x + 15   +   3x - 25   +   4x - 20     =  360

                                                                                 Combine like terms.

13x - 30  =  360

                                                     Add  30  to both sides of the equation.

13x  =  390

                                                     Divide both sides of the equation by  13  .

x  =  30

Since  x  is positive, the largest angle must be  5x + 15 .

( It has the largest coefficient of  x  and all the others either add nothing or subtract something. )

And..... if  x  =  30,       5x + 15  =  5(30) + 15  =  165°

Is this your question?

\(\sqrt{l(l-5)}=6\)

                                                   Square both sides of the equation.

\(\sqrt{l(l-5)}^2=6^2 \\~\\ l(l-5)=36\)

                                                   Distribute the  l  to both terms in parenthesees.

\(l^2-5l=36\)

                                                   Subtract  36  from both sides of the equation.

\(l^2-5l-36=0\)

                                                   Use the quadratic formula to solve for  l  .

\(l=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(-36)}}{2(1)} \\~\\ l=\frac{5\pm\sqrt{25+144}}{2} \\~\\ l=\frac{5\pm13}{2} \\~\\ \begin{array} \ l=\frac{5+13}{2}\qquad &\text{or}&\qquad l=\frac{5-13}{2}\\~\\ l=9&\text{or}&\qquad l=-4 \end{array}\)

2.3 million  is  what percent of 325,602,223

2,300,000  =  (what) / 100    *  325,602,223

                                                                             Divide both sides by  325,602,223

2,300,000 / 325,602,223      =  (what) / 100

                                                                             Multiply both sides by  100  .

230,000,000 / 325,602,223  =  what

0.706  ≈  what

So...

Approx. 0.706  percent  of  325,602,223  is  2.3 million .

This is multiplying a binomial by a binomial to get a polynomial.

    (3a² - 1)(-3a² + 5)                 Multiply each term in the first set of parenthesees by each term in

                                                 the second set of parenthesees and add these products together.

=  (3a²)(-3a²) + (3a²)(5) + (-1)(-3a²) + (-1)(5)

=       -9a⁴      +    15a²    +     3a²     +  -5

=       -9a⁴      +             18a²             -    5