If the indicators are the same but the reasons are different. Lesson "Multiplication and division of powers"
Each arithmetic operation sometimes becomes too cumbersome to write and they try to simplify it. This was once the case with the addition operation. People needed to carry out repeated addition of the same type, for example, to calculate the cost of one hundred Persian carpets, the cost of which is 3 gold coins for each. 3+3+3+…+3 = 300. Due to its cumbersome nature, it was decided to shorten the notation to 3 * 100 = 300. In fact, the notation “three times one hundred” means that you need to take one hundred threes and add them together. Multiplication caught on and gained general popularity. But the world does not stand still, and in the Middle Ages the need arose to carry out repeated multiplication of the same type. I remember an old Indian riddle about a sage who asked for wheat grains in the following quantities as a reward for work done: for the first square of the chessboard he asked for one grain, for the second - two, for the third - four, for the fifth - eight, and so on. This is how the first multiplication of powers appeared, because the number of grains was equal to two to the power of the cell number. For example, on the last cell there would be 2*2*2*...*2 = 2^63 grains, which is equal to a number 18 characters long, which, in fact, is the meaning of the riddle.
The operation of exponentiation caught on quite quickly, and the need to carry out addition, subtraction, division and multiplication of powers also quickly arose. The latter is worth considering in more detail. The formulas for adding powers are simple and easy to remember. In addition, it is very easy to understand where they come from if the power operation is replaced by multiplication. But first you need to understand some basic terminology. The expression a^b (read “a to the power of b”) means that the number a should be multiplied by itself b times, with “a” being called the base of the power, and “b” the power exponent. If the bases of the degrees are the same, then the formulas are derived quite simply. Specific example: find the value of the expression 2^3 * 2^4. To know what should happen, you should find out the answer on the computer before starting the solution. Entering this expression into any online calculator, search engine, typing “multiplying powers with different bases and the same” or a mathematical package, the output will be 128. Now let’s write out this expression: 2^3 = 2*2*2, and 2^4 = 2 *2*2*2. It turns out that 2^3 * 2^4 = 2*2*2*2*2*2*2 = 2^7 = 2^(3+4) . It turns out that the product of powers with the same base is equal to the base raised to a power equal to the sum of the two previous powers.
You might think that this is an accident, but no: any other example can only confirm this rule. Thus, in general, the formula looks like this: a^n * a^m = a^(n+m) . There is also a rule that any number to the zero power is equal to one. Here we should remember the rule of negative powers: a^(-n) = 1 / a^n. That is, if 2^3 = 8, then 2^(-3) = 1/8. Using this rule, you can prove the validity of the equality a^0 = 1: a^0 = a^(n-n) = a^n * a^(-n) = a^(n) * 1/a^(n) , a^ (n) can be reduced and one remains. From here the rule is derived that the quotient of powers with the same bases is equal to this base to a degree equal to the quotient of the dividend and divisor: a^n: a^m = a^(n-m) . Example: simplify the expression 2^3 * 2^5 * 2^(-7) *2^0: 2^(-2) . Multiplication is a commutative operation, therefore, you must first add the multiplication exponents: 2^3 * 2^5 * 2^(-7) *2^0 = 2^(3+5-7+0) = 2^1 =2. Next you need to deal with division by a negative power. It is necessary to subtract the exponent of the divisor from the exponent of the dividend: 2^1: 2^(-2) = 2^(1-(-2)) = 2^(1+2) = 2^3 = 8. It turns out that the operation of dividing by a negative the degree is identical to the operation of multiplication by a similar positive exponent. So the final answer is 8.
There are examples where non-canonical multiplication of powers takes place. Multiplying powers with different bases is often much more difficult, and sometimes even impossible. Some examples of different possible techniques should be given. Example: simplify the expression 3^7 * 9^(-2) * 81^3 * 243^(-2) * 729. Obviously, there is a multiplication of powers with different bases. But it should be noted that all bases are different powers of three. 9 = 3^2.1 = 3^4.3 = 3^5.9 = 3^6. Using the rule (a^n) ^m = a^(n*m) , you should rewrite the expression in a more convenient form: 3^7 * (3^2) ^(-2) * (3^4) ^3 * ( 3^5) ^(-2) * 3^6 = 3^7 * 3^(-4) * 3^(12) * 3^(-10) * 3^6 = 3^(7-4+12 -10+6) = 3^(11) . Answer: 3^11. In cases where there are different bases, the rule a^n * b^n = (a*b) ^n works for equal indicators. For example, 3^3 * 7^3 = 21^3. Otherwise, when the bases and exponents are different, complete multiplication cannot be performed. Sometimes you can partially simplify or resort to the help of computer technology.
Degree formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.
Number c is n-th power of a number a When:
Operations with degrees.
1. By multiplying degrees with the same base, their indicators are added:
a m·a n = a m + n .
2. When dividing degrees with the same base, their exponents are subtracted:
3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:
(abc…) n = a n · b n · c n …
4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:
(a/b) n = a n /b n .
5. Raising a power to a power, the exponents are multiplied:
(a m) n = a m n .
Each formula above is true in the directions from left to right and vice versa.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:
3. When raising a root to a power, it is enough to raise the radical number to this power:
4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:
5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:
A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:
Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.
A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.
The concept of degree in mathematics is introduced in the 7th grade in algebra class. And subsequently, throughout the entire course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic, requiring memorization of values and the ability to count correctly and quickly. To work with degrees faster and better, mathematicians came up with degree properties. They help to reduce large calculations, convert a huge example into a single number to some extent. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the basic properties of the degree, as well as where they are applied.
Properties of degree
We will look at 12 properties of degrees, including properties of degrees with the same bases, and give an example for each property. Each of these properties will help you solve problems with degrees faster, and will also save you from numerous computational errors.
1st property.
Many people very often forget about this property and make mistakes, representing a number to the zero power as zero.
2nd property.
3rd property.
It must be remembered that this property can only be used when multiplying numbers; it does not work with a sum! And we must not forget that this and the following properties apply only to powers with the same bases.
4th property.
If a number in the denominator is raised to a negative power, then when subtracting, the degree of the denominator is taken in parentheses to correctly change the sign in further calculations.
The property only works when dividing, it does not apply when subtracting!
5th property.
6th property.
This property can also be applied in the opposite direction. A unit divided by a number to some extent is that number to the minus power.
7th property.
This property cannot be applied to sum and difference! Raising a sum or difference to a power uses abbreviated multiplication formulas rather than power properties.
8th property.
9th property.
This property works for any fractional power with a numerator equal to one, the formula will be the same, only the power of the root will change depending on the denominator of the power.
This property is also often used in reverse. The root of any power of a number can be represented as this number to the power of one divided by the power of the root. This property is very useful in cases where the root of a number cannot be extracted.
10th property.
This property works not only with square roots and second powers. If the degree of the root and the degree to which this root is raised coincide, then the answer will be a radical expression.
11th property.
You need to be able to see this property in time when solving it in order to save yourself from huge calculations.
12th property.
Each of these properties will come across you more than once in tasks; it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, to make the right decision, it is not enough to know only the properties; you need to practice and incorporate other mathematical knowledge.
Application of degrees and their properties
They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, and equations and examples related to other branches of mathematics are often complicated by powers. Powers help to avoid large and lengthy calculations; powers are easier to abbreviate and calculate. But to work with large powers, or with powers of large numbers, you need to know not only the properties of the power, but also work competently with bases, be able to expand them to make your task easier. For convenience, you should also know the meaning of numbers raised to a power. This will reduce your time when solving, eliminating the need for lengthy calculations.
The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is a power of a number.
Abbreviated multiplication formulas are another example of the use of powers. The properties of degrees cannot be used in them; they are expanded according to special rules, but in each formula of abbreviated multiplication there are invariably degrees.
Degrees are also actively used in physics and computer science. All conversions to the SI system are made using powers, and in the future, when solving problems, the properties of the power are used. In computer science, powers of two are actively used for the convenience of counting and simplifying the perception of numbers. Further calculations for converting units of measurement or calculations of problems, just like in physics, occur using the properties of degrees.
Degrees are also very useful in astronomy, where you rarely see the use of the properties of a degree, but the degrees themselves are actively used to shorten the notation of various quantities and distances.
Degrees are also used in everyday life, when calculating areas, volumes, and distances.
Degrees are used to record very large and very small quantities in any field of science.
Exponential equations and inequalities
Properties of degrees occupy a special place precisely in exponential equations and inequalities. These tasks are very common, both in school courses and in exams. All of them are solved by applying the properties of degree. The unknown is always found in the degree itself, so knowing all the properties, solving such an equation or inequality is not difficult.
In the last video lesson, we learned that the degree of a certain base is an expression that represents the product of the base by itself, taken in an amount equal to the exponent. Let us now study some of the most important properties and operations of powers.
For example, let's multiply two different powers with the same base:
Let's present this work in its entirety:
(2) 3 * (2) 2 = (2)*(2)*(2)*(2)*(2) = 32
Having calculated the value of this expression, we get the number 32. On the other hand, as can be seen from the same example, 32 can be represented as the product of the same base (two), taken 5 times. And indeed, if you count it, then:
Thus, we can confidently conclude that:
(2) 3 * (2) 2 = (2) 5
This rule works successfully for any indicators and any reasons. This property of power multiplication follows from the rule that the meaning of expressions is preserved during transformations in a product. For any base a, the product of two expressions (a)x and (a)y is equal to a(x + y). In other words, when any expressions with the same base are produced, the resulting monomial has a total degree formed by adding the degrees of the first and second expressions.
The presented rule also works great when multiplying several expressions. The main condition is that everyone has the same bases. For example:
(2) 1 * (2) 3 * (2) 4 = (2) 8
It is impossible to add degrees, and indeed to carry out any power-based joint actions with two elements of an expression if their bases are different.
As our video shows, due to the similarity of the processes of multiplication and division, the rules for adding powers in a product are perfectly transferred to the division procedure. Consider this example:
Let's transform the expression term by term into its full form and reduce the same elements in the dividend and divisor:
(2)*(2)*(2)*(2)*(2)*(2) / (2)*(2)*(2)*(2) = (2)(2) = (2) 2 = 4
The end result of this example is not so interesting, because already in the process of solving it it is clear that the value of the expression is equal to the square of two. And it is two that is obtained by subtracting the degree of the second expression from the degree of the first.
To determine the degree of the quotient, it is necessary to subtract the degree of the divisor from the degree of the dividend. The rule works with the same base for all its values and for all natural powers. In the form of abstraction we have:
(a) x / (a) y = (a) x - y
From the rule of dividing identical bases with degrees, the definition for the zero degree follows. Obviously, the following expression looks like:
(a) x / (a) x = (a) (x - x) = (a) 0
On the other hand, if we do the division in a more visual way, we get:
(a) 2 / (a) 2 = (a) (a) / (a) (a) = 1
When reducing all visible elements of a fraction, the expression 1/1 is always obtained, that is, one. Therefore, it is generally accepted that any base raised to the zero power is equal to one:
Regardless of the value of a.
However, it would be absurd if 0 (which still gives 0 for any multiplication) is somehow equal to one, so an expression of the form (0) 0 (zero to the zero power) simply does not make sense, and to formula (a) 0 = 1 add a condition: “if a is not equal to 0.”
Let's solve the exercise. Let's find the value of the expression:
(34) 7 * (34) 4 / (34) 11
Since the base is the same everywhere and equal to 34, the final value will have the same base with a degree (according to the above rules):
In other words:
(34) 7 * (34) 4 / (34) 11 = (34) 0 = 1
Answer: the expression is equal to one.
