February 2018 – Manroop's Blog

Week 4 – Pre Calc 11

manroopt2015 on February 24, 2018February 24, 2018

In the fourth week of Pre-Calculus 11 we continued the absolute value and radicals unit. This week we learned about addition and subtraction of radicals.

Addition and Subtraction of Radicals: To add and subtract radicals, the radicands have to be of the same value. Not only do the radicands have to be the same, but the index also has to be the same. The coefficients in front of the radical are added or subtracted, not the radicand.

Ex. $\sqrt{23}+\sqrt[4]{23}$

The example above cannot be added together because the index’s are not the same. Although, the radicals have the same radicand, the index of the radicals are different.

Ex. $2\sqrt{21}+\sqrt{21}$

$3\sqrt{21}$

The expression above could be added together because the radicand and the index are the same. I added the coefficients of the radical together and not the value inside the radical.

Ex. $\sqrt{20}+8\sqrt{18}-2\sqrt{50}+\sqrt{45}$

$\sqrt{4\times5}+8\sqrt{9\times2}-2\sqrt{25\times2}+\sqrt{9\times5}$

$2\sqrt{5}+24\sqrt{2}-10\sqrt{2}+3\sqrt{5}$

$5\sqrt{5}+14\sqrt{2}$

In the example above, before adding/subtracting the radicals I simplified everything inside the radical. By simplifying the radical first, I was able to tell whether I could add/subtract the radicals together.

How to Multiply Radicals: To multiply radicals you have to multiply the coefficients by the coefficients and multiply the radicand by the radicand.

Ex. $3\sqrt{5}\times7\sqrt{2}$

$21\sqrt{10}$

It’s best to simplify the radical before multiplying, if possible. By simplifying the radical beforehand, it’ll be easier to multiply the radicals together because the numbers will be smaller.

Ex. $2\sqrt{5}(\sqrt{75}-7\sqrt{8})$

$2\sqrt{5}(\sqrt{25\times3}-7\sqrt{4\times2})$

$2\sqrt{5}(5\sqrt{3}-14\sqrt{2})$

$10\sqrt{15}-28\sqrt{10}$

In the example above, I simplified the radicals to make the numbers more manageable to multiply. After simplifying the radicals I began to distribute inside the brackets.

How to Divide Radicals: When dividing a question that involves radicals you can only divide radicals by radicals and coefficients by coefficients.

Ex. $\frac{\sqrt{2}}{4}$

The example above cannot be simplified any further. Although 4 and 2 have a common factor of 2, they cannot be simplified because they are not in the same form.

Ex. $\frac{6\sqrt{10}}{9\sqrt{5}}$

$\frac{2\sqrt{2}}{3\sqrt{1}}$

$\frac{2\sqrt{2}}{3}$

The most important rule for dividing radicals is that a radical cannot be left in the denominator of a fraction. If the denominator does contain a radical you have to rationalize the denominator.

Rationalize the Denominator: To rationalize the denominator with a radical you have to multiply the whole fraction by the denominator.

Ex. $\frac{\sqrt{35}}{\sqrt{10}}$

$\frac{\sqrt{7}}{\sqrt{2}}$

$\frac{\sqrt{7}}{\sqrt{2}}*\frac{\sqrt{2}}{\sqrt{2}}$

$\frac{\sqrt{14}}{\sqrt{4}}$

$\frac{\sqrt{14}}{2}$

When the denominator is a binomial containing a radical you have to multiply by the conjugate. To find the conjugate of a binomial you have to flip the sign in the middle of the binomial. The conjugate of $6+\sqrt{3}$ is $6-\sqrt{3}$ .

Ex. $\frac{\sqrt{5}}{3-2\sqrt{7}}$

$\frac{\sqrt{5}}{3-2\sqrt{7}}\times\frac{3+2\sqrt{7}}{3+2\sqrt{7}}$

$\frac{3\sqrt{5}+2\sqrt{35}}{9+6\sqrt{7}-6\sqrt{7}-4\sqrt{49}}$

$\frac{3\sqrt{5}+2\sqrt{35}}{9-28}$

$\frac{3\sqrt{5}+2\sqrt{35}}{-19}$

Week 3 – Absolute Value & Radicals

manroopt2015 on February 17, 2018February 24, 2018

This week in Pre-Calc 11 we started the Absolute Value and Radicals unit. We learned the basics about absolute value and reviewed simplifying radicals.

What is the Absolute Value of a Real Number? The absolute value of a real number is the principle square root of the square of a number.

Sign: $\mid\mid$

The absolute value of a number is it’s distance away from 0. The absolute value of a number is always positive.

Ex. 1 $\mid-10\mid=10$

Ex. 2 $\mid10\mid=10$

In the examples above I showed the absolute value of a positive and negative number. -10 and 10 have the same absolute value because they are the same distance away from 0; both numbers are 10 spaces away from 0.

How do you simplify an expression? The $\mid\mid$ act like brackets. You simplify everything inside the absolute value sign and then move to the outside.

Ex. $9\mid3+6(-6)\mid$

$9\mid3-36\mid$

$9\mid-33\mid$

$9(33)$

297

In the example above, I solved everything inside the absolute value symbol. Then, I figured out the absolute value -33 which is 33. Lastly, I multiplied the absolute value by the number outside of the absolute value symbol.

How to Simplify Radicals: To simplify radicals you have to break the radicand down by using perfect squares.

Ex. $\sqrt18$

$\sqrt9*2$

$\sqrt9\times\sqrt2$

$3\sqrt2$

In the example I broke down the square root of 18 by using the number 9. 9 can divide into 18 and is a perfect square. After I figured out that $9*2=18$ I removed 9 from inside the radical by finding the square root of 9 and moving it to the outside.

Week 2 – Pre-Calc 11

manroopt2015 on February 10, 2018

This week in Pre-Calc 11 we continued to learn about different series and sequences. This week we learned about Geometric Sequences and Series.

What is a Geometric Sequence? A Geometric Sequence is a group of numbers that is multiplied by a constant ratio.

Ex. 2 ,10 ,50 ,250…

The common ratio for this sequence is $\frac{5}{1}$ or you can represent the common ratio by using the number 5. The common ratio is represented by the letter $r$ . A Geometric Series can NEVER have a dividing pattern. The common ratio of a geometric sequence CANNOT equal to 0 or 1.

Ex. 12, 6, 3, 1.5…

For this pattern $r=\frac{1}{2}$ although at first you might think that the pattern is dividing by 2 each time. In a geometric sequence the numbers have to be MULTIPLIED by a constant ratio each time, so this pattern can be represented as $r=\frac{1}{2}$ because multiplying by $\frac{1}{2}$ is the same as dividing by 2 each time.

How do you find r? The formula to find r is $r=\frac{t_n}{t_(n-1)}$

Ex. $\frac{1}{3},\frac{-1}{6},\frac{1}{12}$

$t_n=\frac{1}{12}$

$t_(n-1)=\frac{-1}{6}$

$r=\frac{t_n}{t_(n-1)}$

$r=\frac{1}{12}\div\frac{-1}{6}$

$r=\frac{1}{12}\times\frac{6}{-1}$

$r=\frac{6}{-12}=\frac{-1}{2}$

$r=\frac{-1}{2}$

How do you find $t_n$ ? The formula for $t_n$ is $t_n=a*r^{n-1}$ . Instead of the first term being represented as $t_1$ , geometric sequences represent the first term as “a”. Similar to arithmetic sequences, geometric sequences also use “n” to represent the number of terms.

Ex. Find $t_{11}$ for the following geometric sequence: 5, 10, 20, 40…

$r=2$

$a=5$

$t_n=a*r^{n-1}$

$t_{11}=5*2^{10}$

$t_{11}=5*1024$

$t_{11}=5120$

What is a Geometric Series? A Geometric Series is similar to a geometric sequence except you replace the commas with a plus sign to find the sum of a certain number of terms.

Ex. $2+4+8+16$