Algebra

Exponents originated in ancient mathematics, with their roots traceable to the work of Babylonian and Egyptian mathematicians who used primitive forms of multiplication. The formal notation for exponents emerged in the 16th century, primarily through the work of European mathematicians like Michael Stifel and later René Descartes. They began using exponents to denote repeated multiplication, allowing for more concise mathematical expressions. This notation laid the groundwork for modern algebra and exponential functions as we know them today.

The mucociliary escalator is a vital defense mechanism of the respiratory system. It consists of cilia on the surface of airway epithelial cells that move in a coordinated manner to transport mucus, which traps inhaled particles, pathogens, and debris, upward towards the throat. This process helps clear the airways and prevents infections, ensuring that the respiratory system remains clear and functional. By facilitating the removal of mucus and contaminants, the mucociliary escalator plays a crucial role in maintaining respiratory health.

The mathematical symbol for "approaches" is typically represented by the arrow "→". This symbol is often used in limits to indicate that a value is getting closer to a certain point, such as in the expression ( x \to a ), which means "x approaches a."

That's called the figure's area.

That's called the figure's area.

That's called the figure's area.

That's called the figure's area.

Is a PYRAMID

NB Do NOT confuse with a TETRAHEDRON, which is three triangles on a triangular base. Thereby making a shape of FOUR sides.

From a common focal point the six triangles for an HEXAGON.

NO!!!

You need to learn a little bit of Latin.

Triangle (Trigon) ; ;Tri = 3

Pentagon ; penta -5

A few other Latin prefixes.

Tetra/Quad ; = 4

Hexa = 6

Hepta = 7

Octa = 8

Nona = 9

Deca = 10

Centa = 100

Oh, dude, there are technically 35 triangles in a pentagon. You've got the main five triangles from the pentagon itself, then you can make 10 more by connecting each vertex to the other non-adjacent vertices, and finally, you can make 20 more by connecting each vertex to the midpoint of the opposite side. So, yeah, 35 triangles in total, if you're into that kind of thing.

Turn it over.

e.g.

1 1/2 = 3/2

It reciprocal is 2/3

As an example.

1 1/2 _.

[(1 x 2) + 1] / 2 =>

[2 +1} / 2 =>

3/2 Improper Fraction.

or

5 7/8

[{5 x 8) + 7] / 8 =>

[40 + 7] / 8

47/8

Similarly for any other mixed number.

nth root (a) = a^(1/n)

mth root a^(1/m)

a^(1/n) + a^(1/m) = a^(1/n) + a^(1/m)

However. when multiplying

a^(1/n) X a^(1/m) = a^([m + n]/[mn])

Think of addition of fractions , where the exponents are concerned.

NB This can only be done when the coefficient 'a' is the same for both numbers.

NNB a^(1/n) means the 'n th root' of 'a'.

To cancel out exponents, you can use the property of exponents that states if you have the same base, you can subtract the exponents. For example, in the expression (a^m \div a^n), you can simplify it to (a^{m-n}). Additionally, if you have an exponent raised to another exponent, such as ((a^m)^n), you can multiply the exponents to simplify it to (a^{m \cdot n}). If you set an expression equal to 1, you can also solve for the exponent directly by taking logarithms.

13.3

7n

-1n,0n,1n,2n,3n,4n,5n,6n,7n

1 2 3 4 5 6 7 8 9

A vacuole is a membrane-bound organelle found in plant and fungal cells, as well as some protists and animal cells. Its primary function is to store nutrients, waste products, and other substances, helping to maintain cellular homeostasis. In plant cells, vacuoles also play a crucial role in maintaining turgor pressure, which keeps the plant rigid and supports its structure. Additionally, vacuoles can be involved in processes such as degradation and recycling of cellular components.

It is an equation. Taking it step by step:

37 - k = 17

-k = 17 - 37

-k = -20

k = 20

An expression where the highest power of ( x ) is 2 is called a quadratic expression. It typically takes the form ( ax^2 + bx + c ), where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). Quadratic expressions can be represented graphically as parabolas.

The conclusion of Boolean Algebra functions is that they can be simplified and manipulated using specific rules and laws, such as the laws of identity, null, idempotent, and De Morgan's theorem. This simplification aids in designing digital circuits and systems, ensuring efficiency and reducing complexity. Ultimately, Boolean Algebra provides a framework for analyzing and implementing logical expressions in computer science and electronics.

To map the field of a variable quantity, one can use techniques such as contour mapping or vector field visualization. First, gather data points representing the variable quantity across a defined area. Then, apply interpolation methods to estimate values between these points, creating a continuous representation. Finally, visualize the results using graphs or software tools that can illustrate the spatial variations in the field.

Nonexamples of slope include horizontal lines, which have a slope of zero, and vertical lines, which have an undefined slope. Additionally, a constant function, such as (y = 5), has no change in (y) regardless of the change in (x), thus demonstrating no slope. Lastly, a flat, level surface, like a tabletop, also does not exhibit slope.

2x^(3)y + 18xy - 10x^(2)y -90y

Rearrange in ascending powers of 'x' .

Hence

2x^(3)y - 10x^(2)y + 18xy -90y

'2' & 'y' are common factor to all four terms.

Hence

2y[ x^(3) - 5x^(2) + 9x - 45]

Inside the 'square' brackets we factor the first two terms and the last two terms.

Hence

2y[(x^(2){x - 5} + 9{x - 5}]

To the two internal terms 'x - 5' is common. Hence

2y[x^(2) + 9)(x - 5)] Fully factored.

NB 'x^(2) + 9 ' = x^(2) + 3^(2) does NOT factor .

Remember two squared terms with a positive(+) between them does NOT factor. However, two squared terms with a negative(-) between them does factor.

x^(2)y^(2) + 2x^(3) y

To factor

'x^(2) is common to both terms.

Hence

x^(2) [ y^(2) + 2xy]

'y' is common to both terms

Hence

x^(2)y[ y + 2x ] Fully factored.

2x - 3y = 2

3x + 2y = 3

Simultaneous Equations.

Arbitrarily decide which letter to eliminate.

In this case 'y'.

To do this we bring the coefficients of 'y' to the same value. . To do this multiply the top equation by '2' and the bottom eq'n by.3.

Hence

4x - 6y = 4

9x + 6y = 9

We now add the two equations. The addition of '-6y + 6y = 0 ', So 'y' is eliminated.

Hence

13x = 13

x = 1

To find 'y' , we substitute 'x = 1' into either equation.

Hence

4(1) - 6y = 4

4 - 6y = 4

-6y = 0

y = 0

So the answer in ( x,y ) form is ( 1,0 ) .

-3 + 3y = 2y + 5

Add '3' to both sides

Hence

3y = 2y + 8

Subtract '2y' from both sides.

y = 8 The answer!!!!!

(3x + 2y)(5x - 3y)

Apply FOIL.

F(First) ; 3x X 5x = 15x^(2)

O(Outside) ; 3x X -3y = - 9xy

I(Inside) ; 2y X 5x = 10xy

L(Last) ; 2y X - 3y = - 6y^(2)

15x^(2) - 9xy + 10xy - 6y^(2)

Collect 'like' terms

15x^(2) + xy - 6y^(2)

The answer!!!!!!