How to calculate the power of the fraction and how to calculate the power of the fraction

The numerator is the power and the denominator.

for the root order. a^(n/m)

The nth power of a is open to the m power.

For example, to the power of (12 7).

First replace it with 2 3 original formula is to square 12 7 first and then open 3 power, the numerator and denominator are separated to make the corresponding square to the 3rd power, and finally do division.

For example, if you are powering 3 to the 5th power of 2, you will first calculate the 3rd power of 2, and then open to the 5th power.

Fractional exponential power.

It is a collective term for the exponential power of positive fractions and the exponential powers of negative fractions.

The exponential power of a fraction is the exponent of a number as a fraction, and the exponential power of a positive number is the radical.

Another representation of . The exponential power of a negative fraction cannot be calculated using a radical formula, but other algorithms are used, which is the focus of high school algebra.

am n = ( am) to the nth power , a>0, m, n z and n >1).

Proof: Order ( AM) to the nth power = b

Take both sides to the nth power, yes.

am = bn

am n= am(1 n) = ( bn)(1 n) = b = am to the nth power.

i.e. am n = ( am ) to the nth power.

Stipulation: The meaning of the exponential power of positive fractions of a positive number is - n n to the m power of a = n a to the m power (a>0, m and n belong to positive integers.

n>1）

A positive fractional exponential power of 0 is equal to 0, and a negative fractional exponential power of 0 does not make sense.

It is pointed out that after the meaning of the exponential power of fractions is specified, the concept of exponents is extended from integer exponents to rational exponents, and the operational properties of integer exponential powers can also be extended to the exponential powers of rational numbers

For any rational number r,s, they all have the following arithmetic properties.

1)ar×as=a(r+s) (a>0,r,s∈q)

2) (ar)s=ars (a>0,r,s∈q)

3) (ab)r=ar×br (a>0,b>0,r∈q)

To calculate the power of a fraction, you can do it by following these steps:

1.First, convert the fraction into an exponential form. For example, express the fraction as a b, where a is the numerator and b is the denominator.

2.Next, the exponential applies to the numerator and denominator exponents. That is, to find powers for a and b respectively.

3.Finally, the power result is re-expressed as a fraction. If the result of a power is an integer, then it can be directly expressed as an exponential form of the molecule. If the result of a power is a decimal or irrational number, it can be reduced to the simplest fraction or expressed as a decimal.

Here's a concrete example:

Evaluate the power (3 4) of the fraction (2 3).

First, convert the fraction to exponential: (2 3) (3 4).

Next, find the powers of the numerator 2 and the denominator 3: 2 (3 4) 3 (3 4).

Finally, the power result is re-expressed as a fraction: (3 to the 4th power of 3 on 2) (3 to the 4th power of 3 on the open).

Note that it can be difficult to do accurate calculations of irrational numbers and decimals when calculating powers, and you can use a calculator or math software to help with the calculations.

The calculation of the power of a fraction can be done with the following steps:

1.Convert the power of fractions to the radical form. For example, convert the power of a fraction into a corresponding radical expression.

2.Calculate the exponential power of the base of the radical. That is, the exponential power of the base is calculated first, and then the root of the resulting result is calculated.

For example, if you want to calculate the power of 1 2 (i.e. the square root of 2), you can do it by following these steps:

1.Convert 1 2 to the root form, i.e. 2.

2.Calculate the exponential power of 2 and the result is 4.

3.Calculate the square root of 4 to get 2.

So, 1 to the power of 2 is equal to 2.

To calculate the power of fractions, we can use the following steps:

1.Write the power of fractions as a radical. For example, write a (m n) as n (a m), where a is the base, m is the numerator, and n is the denominator.

2.Calculate the m power of the base a. That is, the value of a m is calculated.

3.Calculates the value under the root number. The result of a m is taken with an nth root number.

As an example, let's say we want to compute 8 (2 3):

1.Write the power of fractions in the radical form, i.e., 8 (2 3) = 3 (8 2).

2.Calculate the power of base 8 to the power of 2, i.e. 8 2 = 64.

3.Calculate the value under the root number, i.e. 3 64 = 4.

So, 8 (2 3) has a value of 4.

It should be noted that the calculation of fractional powers may involve squares, fractional operations and exponential operations, so in the specific calculation, it is necessary to pay attention to the order and law of each step to ensure the correctness of the calculation.

The operation algorithm for the exponential power of fractions is: (a b) c = a (bc), where a, b, and c are real numbers, and b, c, ≠ 0.

For example, (2 3) (4 5) = 2 = 2 = 2 = (2 4)*(2 8 5) = 16*.

The method of calculating the fractional power of a constant: for example, the 3 and 4 powers of 2 are 4 (root number 2 3).DenominatorThe number of times the root number is done outside, and the molecule is done inside the root number.

A constant fractional power is a name that has a certain meaning and is used in place of a number or string.

Its value never changes. It is often capitalized in mathematics"c"to denote a constant. A mathematical constant.

Refers to a constant with an unchanged value.

The opposite of this is variables. Unlike most physical constants, the definition of mathematical constants is independent of all physical measurements.

Power of order of a constant term:

The order of a monomial is the exponential sum of the letters, and the constant term has no letters, so the order is 0. With regard to the number of constant terms, it can also be understood that if a constant is given a letter factor that is not equal to 0 and has an exponent of 0 (the power of zero of a non-zero is equal to 1), it is obvious that the order of the constant term is 0.

In particular, 0 is also a constant term, but 0 does not have a degree.

There is one more thing to note, and e. Not letters, but constant terms Dachang. For example, the coefficient of ab should not be 1, but should be . Because is a concrete number: it is also a constant term. e = 。

Therefore, the number of constant terms (except 0) is 0.

The power exponent is an operation of a fraction, equal to the power of the numerator to the power of the denominator and the denominator.

For example: a (m n) = n (a m).

i.e. n to the m power of a. It is equal to the m power of a to the nth power.

The power is the fraction equivalent to the square operation.

The numerator of the exponent is used as the exponent and denominator of the number in the root number.

As the root number outside the exponent, that is, open several squares.

This question is to calculate 780 * 3558 first, and then open it square, 4000 is also open square, and divide the two numbers after opening the most clear land.

Both are correct, you are wrong, a 3 4 is a (3 4) (1 2), the number under the root number is a power exponent, m=3, n=4, multiple root numbers are just its exponents are reduced by half, the exponential operation is the addition, subtraction, multiplication and division of the exponent, it should be carried out in parts, and the layers of the stupid chain are peeled off.

The landlord thinks too much, and it is easy to lead the argument to confuse himself.

You might as well put 4000 out of 400 to get 20 by asking for the root, and about 780, to get 39 * (3558 10) = 39 * (3 * 593 5) imitation forest (1 2).

593 is a prime number that can only be reserved or calculated with a calculation and selling Daling calculator.

The rules for calculating the power of fractions are as follows:

1.For any non-zero real number a, its positive integer power can be found by multiplying a by its own n times, where n is the exponent of the power. For example, to the 3rd power of 2, which is 2 times 2 times 2 times 2, the result is 8.

2.For any real number a and any positive integer n, the -n power of a is equal to dividing the reciprocal of a by n power by n times. For example, the -3rd power of 2 is 1 divided by (2 to the 3rd power), multiplied by 3 times, and the result is.

3.For any integer a and any integer n, the nth power of a a is equal to multiplying a by n of itself. For example, 3 to the power of 4 is 3 times 3 times 3 times 3 times 3 times 3, and the result is 81.

4.For any real number a and any real number b (b is not equal to 0), the power of the first quarter of a b is equal to the power b of a to the power b. For example, the power of 3 of 2 is to turn the power of 2 to the power of 3 to the power of 3, and the result is.

Expand your knowledge:

After stipulating the meaning of the exponential power of fractions, the concept of exponential macrogod was extended from integer exponents to rational exponents. Multiply with the power of the base, the base does not change, and the exponents are added; Divide by the power of the same base, the base does not change, and the exponent is subtracted; the power of the power, the base is unchanged, and the exponents are multiplied; Multiply the power of the same exponent, the exponent is unchanged, and the base is multiplied; Divide by the power of the same exponent, the exponent does not change, and the base is divided.

Introducing the exponential power of fractions and generalizing the operation properties of power to the meaning of rational numbers, the operation of power and square is unified into the same operation, that is, the operation of power. For the calculation results, it is not mandatory to express them in a uniform form, and if there are no special requirements, they are generally expressed in the form of fractional exponential powers. However, the result cannot contain both a root and a fractional exponent, nor can it have both a denominator and a negative exponent.

The above are the basic rules for calculating the power of fractions. It is important to note that these rules must be followed when calculating the power of fractions, otherwise it may lead to incorrect results. At the same time, in order to ensure the accuracy of the calculation, you can use a signal loss calculator or a computer program to calculate the power of fractions.

The numerator is the power order and the denominator is the root order. A (n m), a to the n power of the m power.

For example, the power of (12 7) will be replaced by 2 3 original formula is to square 12 7 first and then open to the third power, the numerator and denominator are separated to the corresponding square to the third power, and finally divide again. For example, if you are powering 3 to the 5th power of 2, you will first calculate the 3rd power of 2, and then open to the 5th power.

Extended content

Studying mathematics has many benefits for both individuals and society. The following is a detailed description of the benefits of learning mathematics, including the improvement of thinking skills, the strengthening of problem-solving skills, the cultivation of logical reasoning skills, the development of creativity, and the application of mathematics in daily life and career.

1. Improvement of thinking ability

Mathematics is a logically rigorous subject, and learning mathematics can cultivate people's thinking skills, including logical thinking, analytical thinking, abstract thinking, etc. By solving mathematical problems, people need to observe, reason, generalize, and summarize, and these thought processes can exercise people's brains and improve thinking efficiency.

2. Strengthening problem-solving ability:

Mathematics is a subject that develops problem-solving skills. By learning mathematics, one can master a range of problem-solving methods and skills, such as analyzing problems, formulating solutions, verifying answers, etc. These abilities are useful not only in the field of mathematics, but can also be applied to a variety of problems in other disciplines and in real life.

3. Cultivation of logical reasoning ability

Mathematics is a subject that focuses on logical reasoning. Learning mathematics can cultivate people's logical thinking ability, including observing problems, extracting rules, and deriving conclusions. By solving mathematical problems, people can train their logical way of thinking and improve the accuracy and rigor of their thinking.

4. Development of creativity

While mathematics is considered a rigorous subject, it also encourages students to develop their creativity. In the process of solving mathematical problems, students need to use various methods and strategies to find solutions and discover new mathematical laws and relationships. This kind of creative thinking will have a positive impact on the overall thinking ability and innovation ability of students.

5. Application of mathematics in daily life and occupation:

Mathematics has a wide range of applications in everyday life and careers. In everyday life, mathematics helps us calculate our shopping spending, analyze data, create budgets, and more. In the profession, mathematics is widely used in science, engineering, economics, finance, and other fields, such as physicists using mathematical models to describe natural phenomena, and engineers using mathematics to calculate structures and designs.

To sum up, learning mathematics is important for both individuals and society. Not only does it improve thinking skills, problem-solving skills, and logical reasoning skills, but it also fosters creativity and plays an important role in daily life and career development. Therefore, I encourage you to learn more about mathematics and apply it in real life to enjoy the benefits of mathematics.

The numerator is the power order and the denominator is the root order.

a^(n/m)

The nth power of a is open to the m power.