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.