"Exponent" Exponential function and logarithmic function PPT

Description

"Exponent" Exponential function and logarithmic function PPT

Explanation of curriculum standards

1. Understand the concepts of nth power and radical formulas, and master the properties of radical formulas.

2. Be able to use the properties of radical expressions to perform operations on radical expressions.

3. Understand the meaning of fractional exponent powers, and master the mutual conversion of radicals and fractional exponent powers.

4. Master the operational properties of real exponential powers, and be able to simplify or evaluate algebraic expressions.

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1. nth root

1. We learned square roots and cube roots in junior high school. Are there fourth square roots, fifth square roots,..., nth square roots?

(1) What is a square root? What is a cube root? How many square roots are there in a number? What about cube roots?

Tip: According to the definitions of square roots and cube roots, there are two square roots of positive real numbers, and they are opposites of each other. For example, the square root of 4 is ±2. There is no square root for negative numbers. There is only one cube root of a number. For example, the cube root of -8 is -2. ;The square root and cube root of zero are both zero.

(2) Analogous to the definitions of the square root and cube root of a, how to define the nth square root of a?

Tip: nth root: If xn=a, then x is called the nth root of a, where n>1, and n∈N*.

2. Fill in the blanks:

3. Do it:

Express the following expressions using radical expressions.

(1) It is known that x5=2 019, then x=___________;

(2) It is known that x4=2 019, then x=___________.

2. Radical form

1. (1) By analogy with square roots and cube roots, guess: When n is an even number, how many nth square roots are there of a number? What about when n is an odd number?

Tip: a is a positive number: {■(n" is an odd number, the "n" root of "a" has one, which is "√(n&a)", "@n" is an even number, and the "n" of "a" There are two square roots, which are "±√(n&a) ";" )┤

a is a negative number: {■(n" is an odd number, there is only one "n"-th root of "a", which is "√(n&a)", "@n" is an even number, and the "n"-th root of "a" Does not exist;" )┤

The nth root of zero is zero, recorded as √(n&0)=0.

(2) According to the meaning of nth square root, it can be seen that (√(n&a))n=a is definitely true, then does the equation √(n&a^n)=a definitely hold true?

Tip: It may not be true. Through exploration, we can get: n is an odd number, √(n&a^n)=a; n is an even number, √(n&a^n)=|a|={■(a", " a≥0"," @"-" a"," a<0"." )┤

2. Fill in the blanks

3. Do it

(1) If (√(n&"-" 2))n=-2(n>1, and n∈N*) is meaningful, then n is a __________ number; (fill in "odd" or "even")

(2) If m

Answer: (1) odd (2) n-m

3. Fractional exponent power

1. (1) What are the operational properties of integer exponential powers?

Tips: ①am·an=am+n; ②(am)n=am·n;

③a^m/a^n =am-n(m>n,a≠0);(4)(a·b)m=am·bm.

(2) How are zero exponent powers and negative integer exponent powers defined?

Tip: Regulation: a0=1(a≠0); 00 is meaningless, a-n=1/a^n (a≠0).

inquiry learning

The concept of radical form

Example 1(1) The cube root of 27 is __________; the fourth root of 16 is __________.

(2) It is known that x6=2 019, then x=__________.

(3) If ∜(x+3) is meaningful, then the value range of real number x is __________.

Analysis: (1) The cube root of 27 is ∛27=3, and the 4th square root of 16 is ±∜16=±2.

(2) From the definition of the radical formula, we can get x=±√(6&2" " 019).

(3) To make ∜(x+3) meaningful, x needs to satisfy x+3≥0, that is, x≥-3.

Answer: (1)3　±2　(2)±√(6&2" " 019)　(3)x≥-3

Reflection on two points that should be paid attention to when it comes to radical concept issues

(1) The parity of n determines the number of nth roots;

(2) When n is an odd number, the sign of a determines the sign of the nth root.

way of thinking

Use substitution method to deal with simplification and proof problems in exponential powers

Typical example: It is known that pa3=qb3=rc3, and 1/a+1/b+1/c=1.

Verify: (pa2+qb2+rc2")" ^(1/3)=p^(1/3)+q^(1/3)+r^(1/3).

Analysis: When I see three equations that are equal, I immediately think of assigning intermediate variables, and use the intermediate variables to construct an equation that can use the known values ​​​​in the question stem.

Prove that pa3=qb3=rc3=k,

Then pa2=k/a, qb2=k/b, rc2=k/c; p=k/a^3, q=k/b^3, r=k/c^3.

∴The left side of the equation proved = k/a+k/b+k/c ^(1/3)

= k 1/a+1/b+1/c ^(1/3)=k^(1/3),

The right side of the equation proved =(k/a^3 )^(1/3)+(k/b^3 )^(1/3)+(k/c^3 )^(1/3)

=k^(1/3) (1/a+1/b+1/c)=k^(1/3).

∴(pa2+qb2+rc2")" ^(1/3)=p^(1/3)+q^(1/3)+r^(1/3).

Drills in class

1. Calculate the value of ∛("(" 2"-" π")" ^3 )+√("(" 3"-" π")" ^2 ) as ()

A.5 B.-1

C.2π-5 D.5-2π

Analysis: ∛("(" 2"-" π")" ^3 )+√("(" 3"-" π")" ^2 )=2-π+π-3=-1. Therefore, choose B .

Answer:B

2. Which of the following expressions is correct ()

A.(n/m)^7=n7m^(1/7)

B.√(12&"(-" 3")" ^4 )=∛("-" 3)

C.∜(x^3+y^3 )=(x+y")" ^(3/4)

D.√(∛9) =∛3

Analysis: ∵(n/m)^7=n^7/m^7 =n7m-7, ∴A is wrong;

∵√(12&"(-" 3")" ^4 )=√(12&3^4 )=∛3,∴B is wrong;

∵∜(x^3+y^3 )=(x3+y3")" ^(1/4),∴C is wrong;

∵√(∛9) =√(9^(1/3) )=9^(1/3×1/2)=3^(1/3)=∛3,∴D is correct.

Answer:D

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Update Time: 2024-10-18

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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