Potentiation properties: what are they and exercises

Potentiation corresponds to the multiplication of equal factors, which can be written in a simplified way using a base and an exponent. The base is the factor that repeats and the exponent is the number of repetitions.

To solve problems with potencies it is necessary to know their properties. See below the main properties used in power operations.

1. Multiplication of powers of the same base

In the product of powers of the same base, we must keep the base and add the exponents.

The^m. The^no = the^{m + n}

Example: 2². 2³ = 2²⁺³ = 2⁵ = 32

2. Power division of same base

In the division of powers of the same base we keep the base and subtract the exponents.

The^m: a^no = the^{m - n}

Example: 2⁴: 2² = 2^4-2 = 2² = 4

3. power power

When the base of a power is also a power, we must multiply the exponents.

(The^m)^no = the^m.n

Example: (3²)⁵ = 3^2.5 = 3¹⁰ = 59 049

4. Product Power

When the basis of a power is a product, we raise each factor to the power.

(The. B)^m = the^m. B^m

Example: (2. 3)² = 2². 3² = 4. 9 = 36

5. quotient power

When the basis of a power is a division, we raise each factor to the exponent.

(a/b)^m = the^m/B^no

Example: (2/3)² = 2²/3² = 4/9

6. Quotient power and negative exponent

When the base of a power is a division and the exponent is negative, the base and sign of the exponent are inverted.

(a/b)^-n = (b/a)^no

Example: (2/3)^-2 = (3/2)² = 3²/2² = 9/4

7. negative exponent power

When the sign of a power is negative, we must invert the base to make the exponent positive.

The^-n = 1/a^no, to ≠ 0

Example: (2)^-4 = (1/2)⁴ = 1/16

8. Power with rational exponent

Radiciation is the reverse operation of potentiation. Therefore, we can transform a fractional exponent into a radical.

The^m/n = ^noa^m

Example: 5^1/2 = √5

9. Power with exponent equal to 0

When a power has an exponent equal to 0, the result will be 1.

The⁰ = 1

Example: 4⁰ = 1

10. Power with exponent equal to 1

When a power has an exponent equal to 1, the result will be the base itself.

The¹ = the

Example: 5¹ = 5

11. Negative base power and odd exponent

If a power has a negative base and the exponent is an odd number, then the result is a negative number.

Example: (-2)³ = (-2) x (-2) x (-2) = - 8

12. Negative base power and even exponent

If a power has a negative base and the exponent is an even number, then the result is a positive number.

Example: (-3)² = (-3) x (-3) = + 9

Read more about Potentiation.

Exercises on enhancement properties

question 1

Knowing that the value of 4⁵ is 1024, what is the result of 4⁶?

a) 2 988
b) 4,096
c) 3 184
d) 4,386

See Answer

Correct answer: b) 4,096.

Note that 4⁵ and 4⁶ have the same bases. Therefore, the power 4⁶ it can be rewritten as a product of powers of the same base.

4⁶ = 4⁵. 4¹

How do we know the value of 4⁵ just replace it in the expression and multiply by 4, because power with exponent 1 results in the base itself.

4⁶ = 4⁵. 4¹ = 1024. 4 = 4 096.

question 2

Based on the enhancement properties, which of the sentences below is correct?

a) (x. y)² = x². y²
b) (x + y)² = x² + y²
c) (x - y)² = x² - y²
d) (x + y)⁰ = 0

See Answer

Correct answer: a) (x. y)² = x² . y².

a) In this case we have the power of a product and, therefore, the factors are raised to the exponent.

b) The correct one would be (x + y)² = x² + 2xy + y².

c) The correct one would be (x - y)² = x² - 2xy + y².

d) The correct result would be 1, since every power raised to the zero exponent results in 1.

question 3

Apply the properties of the powers to simplify the following expression.

(2⁵. 2^-4): 2³

See Answer

Correct answer: 1/4.

We start solving the alternative from what is inside the parentheses.

2⁵. 2^-4 is the multiplication of powers of equal bases, so we repeat the base and add the exponents.

2^{5 + (-4)} = 2¹

(2⁵. 2^-4): 2³ = 2¹: 2³

Now the expression has turned into a division of powers on the same basis. So let's repeat the base and subtract the exponents.

2¹: 2³ = 2^1-3 = 2^-2

Since the result is a negative exponent power, we must invert the base and sign of the exponent.

2^-2 = (1/2)²

When the potency is based on a quotient, we can raise each term to the exponent.

1²/2² = 1/4

Therefore, (2⁵. 2^-4): 2³ = 1/4.

Get more knowledge with the contents:

• Radiation
• Potentiation Exercises
• Radiation Exercises
• Difference between Potentiation and Radiation