Simplifying exponents

To understand simplification of exponents we first need to establish the rules of exponents.

Exponent rules:
1. b^m * b^n = b^(m+n)
2. b^m/b^n = b^(m-n)
3.(b^m)^n = b^mn
4.v(n&b^m ) = b^(m/n)

Exponentiation:

We know that multiplication corresponds to repeated addition. In the same way, exponentiation corresponds to repeated multiplication. In other words, exponentiation refers to the process of repeated multiplication. For example we can write, 4*4*4 as 4^3 or 5*5*5*5 = 5^4 etc. In general terms, b*b*b*b…. n times = b^n. Here, b is the base and n is called the exponent or the index.

b^2 is usually read as b squared. b^3 is read as b cubed; where as b^4 is read as ‘b raised to power 4’. In the same way b^(any other number) is read as ‘b raised to the power ______’.

Properties of exponents:
1. Exponent can be any real number.
2. When exponent is zero, the value of the term becomes equal to 1. That is to say that b^0 = 1
3. Exponent of one results in the base itself. So, b^1 = b.
4. ?(b?^(n)m) is not the same as b^(n^m ). ?(b?^(n)m) = b^mn where as b^(n^m ) = b^n^m.
5. When exponent is negative it is same as the positive exponent of the reciprocal of base. So, b^(-n) = (1/b)^n.

Rational exponents:

We saw above that exponent can be any real number. But for now we shall look at rational exponents only. A rational exponent would be of the type m/n. Therefore the number with rational exponent would look like this : b^(m/n). Based on the rules of exponents that we saw earlier, we can say that, b^(m/n) = v(n&b^m ). In other words it’s the nth root of b raised to power m. A number with a rational exponent may or may not itself be a rational number. For example, 4^(8/4) = 4^2 = 16. However, 3^(5/2) can be written as v(2&3^5 ) = v(3^4 * 3^1) = 3^(4/2) * 3^(1/2) = 3^2 * 3^(1/2) = 9 * v(3) = 9v(3) is an irrational number.

Solved examples:

1. Simplify: x^6 * x^5
Solution: x^6 * x^5 = x^(6+5) = x^11

2. Simplify: t^10/t^8
Solution: t^10/t^8 = t^(10-8) = t^2

3. Simplify: 5x^3/3x^5
Solution: 5x^3/3x^5 = (5/3)*(x^3/x^5) = (5/3) * (x^(3-5)) = (5/3) * x^(-2) = 5/3x^2

4. Simplify: (125x^2y^3z^2)^0
Solution: (125x^2y^3z^2)^0  = 1. That is because when exponent is zero, the term becomes = 1

What is a ratio

What is a ratio? A ratio is a comparison of two quantities by division. A ratio actually compares one thing happening to another. Ratios compares two or more amounts and are often expressed as fractions in simplest form or as decimals. Ratio can be written in three different ways:-

Example of a ratio

1. Using a colon like 3:2
2. Using a fraction like 3/2
3. Using the word “ to” like 3 to 2

Ratio Example: - Sam has 4 apples and 9 bananas. What is the ratio of his apples to bananas?
Solution: - Total numbers of apples are 4 and bananas are 9. SO the ratio will be 4:9.

Multiplying and dividing any ratio by same non- zero number makes no difference to the ratio. For example, the ratio 3:9 is equal to 1:3.
Two ratios that name the same number are equivalent ratios.  Equivalent ratios can be calculated by multiplying or dividing each term of the ratio by the same non - zero number.

For example: - If we have a ratio 2:7, multiplying both terms by 3 gives a new ratio 6:21
Hence, 2:7 and 6:21 are equivalent ratios.
If we have 12:3, if we reduce this fraction, we get 4:1
Hence 12:3 and 4:1 are equivalent ratios.
Now let us learn about ratio and rates. A rate is a ratio involving two quantities in different units. The rate 225 heartbeat/ 3minutes compare heartbeat to minutes. If the denominator is 1 then it is known as the unit rate. Therefore,
225 heartbeat/3 minutes = 75 heartbeat / 1 minute.
Hence, the unit rate is 75 heartbeats per minute.

Equivalent rates have the same value but use different measurements. We use equivalent rates to help us with unit conversions.

For example: - A jet flies 540 miles per hour. What is the rate in miles per minute?
Solution: - 540 miles/ 1 hour = 1hour/60 min
After simplifying, we get 9 miles per minute.

Bias

Let's learn about bias statistics in today's post.

There are four different types of bias in statistics:

• Spectrum bias
• Omitted variable bias
• Systematic bias
• Cognitive bias

Next time i will help you with the concept of statistics formulas.

Also you can avail help from expert online tutors. Not just in statistics but from an algebra tutor as well.

Do post your comments.

definition of mathematics

Let us learn about definition of mathematics

Mathematics is a study of relationships using different numbers & the study of the relationships between numbers, shapes, & quantities. Mathematics uses signs, symbols, & proofs & includes arithmetic, calculus, algebra, geometry, & trigonometry.

Mathematics is referred as Mathematical operations & processes involved in the solution of a problem or study of some scientific field

Mathematics is a group of related sciences, including geometry, algebra, & calculus, concerned with the study of number, quantity, shape, & space & their interrelationships by using a specialized notation

Mathematics is a science or group of related sciences that deals with the logic of quantity & shape & arrangement

The mathematics is deductive study of shape, quantity, & dependence.

If you are interested to take English Grammar Test, you can click on given link.

In our next blog we shall learn about energy changes I hope the above explanation was useful.Keep reading and leave your comments.

inequalities calculator

Let us learn how to use inequalities calculator

Add all numbers in left side from the in equality.

Then add all numbers in right side from the inequality.

Add the entire variable with coefficients on the left side from inequality.

Add the entire variable with coefficients on the right side from the inequality.

Subtract all number on left side from both sides of the inequality.

Add all number on the left side from both sides of the inequality

Subtract all variable with a coefficient on the right hand side of the inequality from both sides of the inequality

Add all variable with a coefficient on the right hand side of the inequality from both sides of the inequality.

Simplify by dividing on both sides or by multiplying on both side with reciprocal of inequality.

In our next blog we shall learn about opposite words in english I hope the above explanation was useful.Keep reading and leave your comments.

sin 30 degrees

Let us learn about sin 30 degrees

Value of 30 Degree

• The value of sin 30 degrees = ½
• The value of cos 30 degrees =√3/2
• The value of tan 30 degree = 1/√3

zero degree angle is known as an acute angle, because the angle of 2 rays Is less than 90 degree & greater than 0 degree.

sin(135) = sqrt(2)/2
cos(135) = -sqrt(2)/2
sin(30) = 1/2
cos(30) = sqrt(3)/2

Note 135 deg is just the 45 deg angle in the upper left quadrant of the unit circle, where sines are postiive & cosines are negative

sin(135-30) = sin135cos30 - cos135sin30
................... = [sqrt(2)/2][sqrt(3)/2] - [-sqrt(2)/2][1/2]
.................. = sqrt(6)/4 + sqrt(2)/4
.................. = [sqrt(6) + sqrt(2)]/4
................... = 0.9659

sin(135) - cos(30) = sqrt(2)/2 - sqrt(3)/2
................................= [sqrt(2) - sqrt(3)]/2
................................= -0.1589

In our next blog we shall learn about hydraulic brake system I hope the above explanation was useful.Keep reading and leave your comments.

factoring calculator online

Let us solve problem using factoring calculator online

Factor the given expression, 2x² + 4x
Solution:
= 2x² + 4x
Taking the common terms outside,
= 2x(x + 2)
The solution is 2x (x+2).
Solve 6x² + 18x - 24 = 0 for x.
Solution
Given equation is 6x²+18x-24=0 for x.
Solve the given equation we get
. (6x - 24)(x + 1)=0
Here by equating the values to zero
6x-24=0, x+1=0
Here keep constant terms on 1 side & move the variables to other side.
6x=24, x=-1
The final solution is X=3/2, x = -1.

In our next blog we shall learn about hydroxide formula I hope the above explanation was useful.Keep reading and leave your comments.