POWERS AND EXPONENTS

Powers and exponents

Math7th grade LAF

Introduction

Why do you use an exponent?

Students: please read together and loudly the lecture:

Do you know the answer to the question? Hint: The English alphabet has 26 letters.

Solution:If you have to do 3-letter acronyms, this means that each letter of the English alphabet can combine with the other 25 letters. So, if you have 26 letters and you have to combine all these letters three times to make the 3-letter acronyms, the answer is:

26 x 26 x 26 = ______ is the number of 3-letter acronyms you can do.

Imagine you have to multiply a quantity by itself a lot of times.For example: 2 x 2 x 2 x 2 x 2 It is easy to solve, but it is very long to write it down.

-Don't worry, there is a simplified way to write the previous multiplication. You can use a base, an exponent and a power.

This is how you write the simplified way of the previous multiplication. The number 2 (the big number) is the quantity that is multiplied many times by itself. This number is known as the BASE. The number 5 (the small number above and right of the base) is the EXPONENT. It is 5 because it is the number of times the base is multiplied by itself. The power is the value of the exponent. In this case, it is the FIFTH power.

power vs exponent

What is the difference between them?

An exponent expresses the number of times the base is multiplied by itself.The POWER is the value of the exponent.

an exponent is a positive or negative or zero number placed above and to the right of the base.

Example:

Example:

The power is the value of the exponent.If the exponent is 3, this means the power is THIRD or CUBED.

Which is the exponent here? Look at the definition above.

more examples

24 = 2 x 2 x 2 x 2 = 16

It is read as 2 to the fourth power

32 = 3 x 3 = 9

It is read as 3 to the second power (or 3 squared)

53 = 5 x 5 x 5 = 125

It is read as 5 to the third power (or 5 cubed)

61 = 6

It is read as 6 to the first power

50 = 1

It is read as 5 to the zero power

NOTE:

Evaluating exponents

With constants (numerical values) everything's ok, but now what happens if you have variables (letters)?

Now, I will give you the value of "p". p = 2. We just substitute this value in the variable and we will find the answer.

a variable is a symbol for a number where the value is unknown. LEtters are used: x, y, z, a, b, c, etc.

p6 = 26 = 2 x 2 x 2 x 2 x 2 x 2 = 64

Now, what happens if we have two different variables? For example: b2 · a

What happens if you multiply the same variable many times? For example:p · p · p · p · p · p

We know: a = 3 and b = 2. So, we just substitute this values and answer the multiplication:22 · 3 = 4 · 3 = 12

"p" is a variable (we don't know its value yet) but it's been multiplied by itself 6 times, so to simplify it, we write it as: p6.

more examples

a · a · a · a where a = 4

a4 = 44 = 256

2 · 2 · 2 · t · t where t = 5

23 · t2 = 23 · 52 = 8 · 25 = 200

9x where x = 2

92 = 81

x2 + y2 + z2 x= 1, y = 2, z = 3

12 + 22 + 32 = 1 + 4 + 9 = 14

3 · 3 · p · q · q · q p = 4, q = 1

32 · p · q3 = 9 · 4 · 13 = 9 · 4 · 1 = 36

Negative bases

What happens when the negative sign is present in the bases?

Negative base with a positive EVEN exponent.

the bases can also be negative numbers. it is important to write it between parenthesis.

When the negative base has an EVEN exponent, the solution is POSITIVE.

Negative base with a positive ODD exponent.

What happens if you don't write the negative base between parenthesis for the expressions with positive EVEN exponents?You won't get the same answer because its sign will be different. (-4)2 = 16 -42 = -16

When the negative base has an ODD exponent, the solution is NEGATIVE.

Paso 1

perfect squares and perfect cubes

Remember: Squares are the bases with the exponent and power of 2, and the Cubes are the bases with the exponent and power of 3.

A perfect square is the solution of the square of a WHOLE base number (or the solution when the WHOLE base number has an exponent and a power of 2).

Examples:

A perfect cube is the solution of the cube of a WHOLE base number (or the solution when the WHOLE base number has an exponent and a power of 3).

Examples:

multiplication and division of EXPONENTS

How to multiply and divide equal or different bases with exponents?

when you multiply the exponents, the bases may be the same or different, and the powers may be the same or different too.

Multiplication of exponents with the SAME BASES

1. Simply keep the same base number and make the addition of the exponents.

Examples: 23 x 25 = 28 Addition of the exponents: 3 + 5 = 8 32 x 35 = 37 Addition of the exponents: 2 + 5 = 7 54 x 57 = 511 Addition of the exponents: 4 + 7 = 11

Multiplication of exponents with DIFFERENT bases and SAME powers

Multiplication of DIFFERENT bases and DIFFERENT powers

1. First, multiply the bases (this multiplication must be written in parenthesis) and the answer is raised to the power that is equal in both bases.

Examples: 13 x 23 = (1 x 2)3 = 23 = 8 23 x 33 = (2 x 3)3 = 63 = 216 42 x 52 = (4 x 5)2 = (20)2 = 400

1. Solve independently each base with its corresponding exponent, and multiply the solutions.

Examples: 23 x 15 = 8 x 1 = 8 23 = 8 and 15 = 1 32 x 25 = 9 x 32 = 288 32 = 9 and 25 = 32 54 x 32 = 625 x 9 = 5625 54 = 625 and 32 = 9

when you divide the exponents, the bases may be the same or different, and the powers may be the same or different too.

Division of exponents

*with the same base: *with different base and same power: *with different base and different power: That will be explained apart in the next slides.

Division of exponents with DIFFERENT bases and SAME powers

Division of exponents with the same BASES.

1. Solve the division of the two different bases (write this division between parenthesis) and raise the solution to the power (that is equal).

Examples: 82 / 42 = (8/4)2 = 22 53 / 43 = (5/4)3 34 / 24 = (3/2)4

1. Simply keep the same base number and make the substraction of the exponents.

Examples: 25 / 23 = 22 Substraction of the exponents: 5 - 3 = 2 35 / 32 = 33 Substraction of the exponents: 5 - 2 = 3 57 / 54 = 53 Substraction of the exponents: 7 - 4 = 3

Division of exponents with DIFFERENT bases and DIFFERENT powers.

1. In this type of division of exponents, it is necessary to solve each base raised to its power, and at the end, make the division.

Examples: 75/32 = 16807 / 9 23 / 82 = 8 / 64 = 1 / 8

Power to the power rule

When a base is raised to a power and the whole expression is again raised to another power.

The Power to the power rule works like this:

EXAMPLES:

The exponent that is next to the base is multiplied times the exponent that is outside of the parenthesis.

POWERS OF MONoMIALS

Power of the power, multiplication of exponents, coefficients and variables... everything is together!

MONOMIALS: expressions with a bunch of variables and coefficients (numbers) multiplied together, with no addition or subtraction.

Example of a monomial:

Everything that is inside of the parenthesis will be raised to the power from the outside (if something from the inside has an exponent, apply the power to the power rule) = 23x3y12z6

NOTE

nEGATIVE EXPONENTS

How to solve the bases with negative exponents?

A negative exponent is defined as the multiplicative inverse of the base, raised to the power which is of the opposite sign of the given power.

The solution of the previous example is then written like this: (1/3)2 = 1/9.

They are read the same as the numbers with positive exponents but just add the word "negative". Example: 3 to the negative second power or 3 to the power of negative 2.

In simple words, we write the reciprocal of the base and the exponent is written with the opposite sign (it is converted into positive)

For example: 3-2. The base is 3. What is the reciprocal of 3? It is 1/3. So, 1/3 is multiplied 2 times. 1/3 x 1/3 = 1/9.

more examples

It is read as 2 to the negative fourth power

2-4 = (1/2)4 = 1/2 x 1/2 x 1/2 x 1/2 = 1/16

3-2 = (1/3)2 = 1/3 x 1/3 = 1/9

It is read as 3 to the negative second power

5-3 = (1/5)3 = 1/5 x 1/5 x 1/5 = 1/125

It is read as 5 to the negative third power

6-1 = (1/6)1 = 1/6

It is read as 6 to the negative first power

NOTE:

exponent rules

Let's make a summary of all the exponent rules you have learned!

Here is the list of the exponent rules with their respective names.

To multiply two expressions with the same base, make the addition of their exponents while keeping the base the same.

To divide two expressions with the same base, subtract their exponents while keeping the base the same.

When we have a single base with two exponents (one over the other), just multiply the exponents.

Write it and solve it as a multiplication of 2 different bases with the same exponent.

Write it and solve it as a division of 2 different bases with the same exponent.

Any number (except 0) that is raised to the power of zero, is equal to 1.

The expression is transferred from the numerator to the denominator with the change in sign of the exponent values (Write the reciprocal expression).

Inside of the root, write the base raised to the numerator exponent, and outside of the root is the denominator of the exponent, and solve the root.