Powers and Roots

One of the mathematics subject matter for grade 9 in the 2013 revised 2018 curriculum is about “Numbers to Powers and Root Forms”. In this post, we will discuss this in detail, starting from numbers to the power of integers, the form of roots and numbers to the power of fractions. Both of these materials will be discussed in detail in a method that is easy for you to understand.

Power of Integer

The material for exponents and the first form of roots is about numbers to powers of integers. What does it mean? And how to solve the problem?

Definition of Power of Numbers

Power is a mathematical operation for the repeated multiplication of a number as much as its power. The power of a number is a number that is written lower and is slightly upwards. Based on the semantics of writing letters are called superscripts, for example: 2², 3², 4³, and others.

Numbers to exponents can be obtained by repeated multiplication of the same factors.

Power of Integer

Integers consist of integers with positive values, integers with negative values, and zero. So it can be concluded that the numbers to the power of integers are numbers that have positive, negative, and zero powers.

1. Power of 0

For integers to the power of 0, the result is 1. So, any integer whether it is negative or positive, if raised to the power of 0 then the result is 1, but this does not apply to integers 0.

To prove n0 = 1, we can use the properties of the power operation number (2), namely division of numbers to the power of:
n ^a  : n ^b  = n ^a-b  or if reversed
n ^a-b  = n ^a  : n ^b .
If n 0 and a=b, then:
n ^a-b  = n ^a  : n ^b
n ^a-a  = n ^a  : n ^a  ; because aa = 0 and n ^a  : n ^a  = 1, then
n = 1 (proven)

2. Positive Integers

Some of the properties of positive integers, including the following:

• a ^m x a ⁿ  = a ^m+n
• a ^m  : a ⁿ  = a ^{m-n ,}  for m>n and b 0
• (a ^m ) ⁿ  = a ^mn
• (ab) ^m  = a ^m  b ^m
• (a/b) ^m  = a ^m /b ^m  , for b 0

3. Negative Integers

The properties of negative integers are:

If a∈R, a 0, and n are negative integers ,  then:

a ^-n  = 1/a ⁿ  or a ⁿ  = 1/ a ^-n

Counting Operations Involving Numbers to Powers

There are several things that must be considered before you work on arithmetic operations involving numbers to exponents, including:

1. Do the operations in brackets first
2. Continue with power operations
3. Perform multiplication and division operations
4. Perform addition or subtraction operations.

Okay, now we will go into the discussion of arithmetic operations that exist on exponents. There are 2 things that will be discussed, namely multiplication and division.

Multiplication in Powers

In multiplication counting operations in exponents, the following properties apply:

a m  xa n  = a m+n

To better understand multiplication to powers, consider the following example:

6 ³  x 6 ²  = (6 x 6 x 6) x (6 x 6)

6 ³  x 6 ²  = 6 x 6 x 6 x 6 x 6

6 ³  x 6 ²  = 6 ⁵

So we can conclude to be 6 ³  x 6 ²  = 6 ²⁺³  = 6 ⁵

However, there is an exception in the case of a negative prime number. There are a few points you should know:

Negative numbers with even powers	= The result is positive
Odd power negative number	= The result is negative

Division to Powers

For arithmetic division operations on exponents, the following properties will apply:

a m  : a n  = a m-n

In order to understand the above understanding, the following is an example of the question:

6 ⁶  x 6 ³  = (6 x 6 x 6 x 6 x 6 x 6) x (6 x 6 x 6)

6 ⁶  x 6 ³  = 6 x 6 x 6 ((6 x 6 x 6) x (6 x 6 x 6))

6 ⁶  x 6 ³  = 6 ³

So, we can conclude to be 6 ⁶  x 6 ³  = 6 ^6-3  = 6 ³

Simple Rank Form

If there is a simple power equation a ^f(x)  = a ⁿ  where a R which is not equal to 0, then to solve the problem, the left side and the right side must be equated. If you don’t understand, consider the following example:

Find the solution set for the following equations!

3 ^{1 + x}  = 81

Answer:

3 ^{1 + x}        = 81

3 ^{1 + x}        = 3 ⁴

1 + x = 4

x = 4 – 1 = 3

So, HP = {3}.

Forms of Roots and Numbers to Fractions

Now we will discuss the form of roots and what if the numbers are integers but have exponents in the form of fractions? Can this problem be solved? Is it the same way with integer powers and ordinary integer powers?

Withdrawing the Root

The square root is the opposite of the square root. The square root (square root) is denoted by the sign .

9 ²  = 81 means 81 = 9

The square root of a number can be found in the following way.

625 = …

• • Separate the two numbers on the right with a dot to 6.25.

• Find the largest root of the number to the left of point (6), which is 2.
• 2 ² = 4, the number 4 is written under the number 6 and then subtracted, i.e. 6 – 4 = 2.
• Lowering the number 25 completes the remaining 2 to 2.25.
• The result of drawing the roots earlier (2) multiply 2 to 4.
• Find the number n that satisfies 4n × n so that the product is 225 or the largest number under 225. In the example the appropriate value of n is 5, so 45 × 5 = 225
• The number 5 is placed to complete the 2 results of drawing the roots to 25.
• Because 225 – 225 = 0 then 25 is the final result of drawing the square root. If the result of the subtraction is not zero, then do the next number reduction as in steps 4 and 5. So, 625 = 25.

Rational Numbers

Rational powers are a form of fractional exponents. Ratio is comparison. So, the exponent is a fraction.

The rational exponent has the same value as the root form.

The following are the ranking rules:

Properties of Operations on Rational Numbers

For a and b real numbers, b≠0 and m,n are valid rational numbers:

Root Shape

Basically, the properties that have been possessed by exponents are also owned by numbers in the form of roots, namely:

For real numbers a, b and n, m rational numbers of the form n=p/q and m=s/t where p, q, s, t natural numbers apply:

where a and b are not negative when p or s are even.

Characteristics of Root Form

For a, b, c, and d real numbers, apply:
1. Addition and Subtraction of the form of roots

2. Multiplication and division of roots

Algebraic Operations of Root Forms

The most common algebraic operations are addition, subtraction, multiplication, and division. The discussion is as follows:

a. Addition and Subtraction of Root Forms

The formula for the addition of the root form:

a√c + b√c = (a + b) c

The formula for subtracting the root form:

a√c – b√c = (a – b) c

b. Multiplication Operation

For each a and b are positive rational numbers, then the formula that applies is:

ax b = axb

c. Division Operation

For each a, b, p, and q are positive rational numbers, the formula that applies is:

(p√a)/(q√b)= p/q (a/b)

Rationalize the Denominator of the Root Form

How to rationalize the denominator of a fraction with a root form can be categorized into several categories. Among others are:

a. Fraction form  a/√b

In the fraction  a/√b  there is a rational number a and the root form of  b  is to make a multiplication between  b /√b with  the fraction. Later, the multiplication form of the root form will be like this:

b. Fractional form or  c/a-√b  or  c/a+√b

How to rationalize the next root form is related to the product pair (a – b) and (a + b), where the rational numbers are a and b and the root form is b. These two product pairs can be solved with a distributive property such as (a + b)( a – b) = a² – a√b + a√b – b = a² – b.

The number (a + b) multiplied by (a – b) gives a rational number. In this case (a – b) is a compound of (a + b) and vice versa or (a – b) and (a + √b) are examples of a group of roots. For example 3 – 2 mate with 3 + 2 and 5 + 3 mate with 5 – 3.

For how to rationalize a fraction with this form the roots can be like this:

That was the discussion for junior high school mathematics about numbers with exponents and roots for grade 9. Hopefully it’s useful!