Mathematical Relationships and Basic Graphs

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Math › Mathematical Relationships and Basic Graphs

Questions 1 - 10

Add the fractions: $\frac{5}{9}+\frac{7}{18}$

$\frac{17}{18}$

$\frac{17}{9}$

$\frac{2}{3}$

$\frac{5}{6}$

$\frac{8}{9}$

Explanation

In order to add the fractions, change the denominator of the first fraction so that both denominators are common.

$\frac{5(2)}{9(2)}+\frac{7}{18}$

Simplify the fractions.

$= \frac{10}{18}+\frac{7}{18} = \frac{17}{18}$

The answer is: $\frac{17}{18}$

$\sqrt{40}+2\sqrt{160}+5\sqrt{90}$

$25\sqrt{10}$

$8\sqrt{290}$

$7\sqrt{290}$

$8\sqrt{10}$

$9\sqrt{10}$

Explanation

Adding and subtracting radicals cannot be done without having the same number under the same type of radical. These numbers first need to be simplified so that they have the same number under the radical before adding the coefficients. Look for perfect squares that divide into the number under the radical because those can be simplified.

$\sqrt{4\cdot 10}+2\sqrt{16\cdot 10}+5\sqrt{9\cdot 10}$

Now take the square root of the perfect squares. Note that when the numbers come out of the square root they multiply with any coefficients outside that radical.

$2\sqrt{10}+2\cdot4 \sqrt{10}+5\cdot 3\sqrt{10}$

$2\sqrt{10}+8 \sqrt{10}+15\sqrt{10}$

Since all the terms have the same radical, now their coefficients can be added

$25\sqrt{10}$

Add the fractions: $\frac{3}{4}+\frac{8}{3}$

$\frac{41}{12}$

$\frac{21}{12}$

$\frac{31}{12}$

$\frac{38}{12}$

Explanation

Find the least common denominator to these fractions.

Multiply both denominators together.

$4\times3=12$

Convert the fractions using this denominator.

$\frac{3}{4}+\frac{8}{3} = \frac{9}{12}+\frac{32}{12} = \frac{41}{12}$

The answer is: $\frac{41}{12}$

Evaluate the infinite series for

Explanation

The first term of this sequence is 10. To find the common ratio r, we can just divide the second term by the first: $-9 \div 10 = - 0.9$ . So "r" is -0.9. We can find the infinite sum using the formula $\frac{a}{1-r}$ where a is the first term and r is the common ratio:

$\frac{10}{1-(-0.9)} = \frac{10}{1+0.9 } = \frac{10}{1.9} \approx 5.263$

What is the value of ?

Explanation

! is the symbol for factorial, which means the product of the whole numbers less than the given number.

Thus, .

Simplify.

$\sqrt{5}\cdot\sqrt{2}$

$\sqrt{10}$

$\sqrt{7}$

$2\sqrt{5}$

$5\sqrt{2}$

Explanation

When multiplying radicals, you can combine them and multiply the numbers inside the radical.

$\sqrt{5}\cdot\sqrt{2}=\sqrt{5\cdot2}=\sqrt{10}$

Solve the equation: $\sqrt{8x-3} = 30$

$\frac{903}{8}$

$\frac{897}{8}$

$\frac{903}{4}$

$\frac{57}{8}$

$\frac{63}{8}$

Explanation

$(\sqrt{8x-3}) ^2= 30^2$

Add three on both sides.

Divide by 8 on both sides.

$\frac{8x}{8}=\frac{903}{8}$

The answer is: $\frac{903}{8}$

Add the radicals, if possible: $\sqrt{18}+\sqrt{50}+\sqrt{200}$

$18 \sqrt{2}$

$24 \sqrt{2}$

$9 \sqrt{3}$

$6\sqrt3$

$\textup{The radicals cannot be added.}$

Explanation

Simplify all the radicals to their simplest forms. Use the perfect squares as the factors.

$\sqrt{18} =\sqrt{9\times 2}= \sqrt{9}\cdot \sqrt{2} = 3\sqrt{2}$

$\sqrt{50}= \sqrt{25}\cdot \sqrt{2} = 5 \sqrt{2}$

$\sqrt{200} = \sqrt{100}\cdot \sqrt{2} = 10\sqrt2$

Add the like terms together.

$3 \sqrt{2}+5 \sqrt{2}+10 \sqrt{2} = 18 \sqrt{2}$

The answer is: $18 \sqrt{2}$

Simplify the following expression:

$13x^{24}$

$40x^{24}$

Explanation

Simplify the following expression:

When multiplying exponents, we need to recall the rule. When we multiply exponents of like base together, we add the exponents. This is seperate from the numbers out in front (the 5 and the 8) those two numbers will be multiplied as normal.

We can sort of rearrange the expression to get the following:

$5x^6*8x^3=(5*8)(x^6*x^3)=40x^{6+3}=40x^9$

Making our answer:

Determine the sum, if possible: $2-\frac{1}{3}+\frac{1}{18}-...$

$\frac{12}{7}$

$-\frac{1}{6}$

$\textup{The series will diverge.}$

$\frac{8}{5}$

$\frac{5}{3}$

Explanation

Write the formula for the sum of an infinite series.

$\textup{Sum} =\frac{\textup{first term}}{1-r}$

The first term is two. To determine the common ratio, we will need to divide the second term by the first, third term with the second, and so forth. The common ratio should be same for each term.

$r=\frac{-\frac{1}{3}}{2} = -\frac{1}{3}\div 2 = -\frac{1}{3} \times \frac{1}{2} = -\frac{1}{6}$

$r=\frac{\frac{1}{18}}{-\frac{1}{3}} = \frac{1}{18}\div -\frac{1}{3} = \frac{1}{18}\times -\frac{3}{1} =-\frac{1}{6}$

The common ratios are verified to be the same. Substitute the into the formula. This value must be between negative one and one or the series will diverge!