The rules of exponentiation are rules that help us perform operations like addition, subtraction, multiplication, and division with powers. In certain exercises, if the rules of exponentiation are not used correctly, it will be very difficult to find the solution, so it's important to know them.

Don't worry! These aren't complicated rules. If you make an effort to understand them and practice enough, you'll be able to apply them easily.

Understanding Exponents

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According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:$a^n\times a^m=a^{n+m}$

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

$7^{9+1}=7^{10}$

Answer

$7^{10}$

Exercise #2

$8^2\cdot8^3\cdot8^5=$

Video Solution

Step-by-Step Solution

All bases are equal and therefore the exponents can be added together.

$8^2\cdot8^3\cdot8^5=8^{10}$

Answer

$8^{10}$

Exercise #3

$\frac{1}{12^3}=\text{?}$

Video Solution

Step-by-Step Solution

First, we recall the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression we obtained:

$\frac{1}{12^3}=12^{-3}$Therefore, the correct answer is option A.

Answer

$12^{-3}$

Exercise #4

$\frac{81}{3^2}=$

Video Solution

Step-by-Step Solution

First, we recognize that 81 is a power of the number 3, which means that:

$3^4=81$We replace in the problem:

$\frac{81}{3^2}=\frac{3^4}{3^2}$Keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{3^4}{3^2}=3^{4-2}=3^2$Therefore, the correct answer is option b.

Answer

$3^2$

Exercise #5

$\frac{2^4}{2^3}=$

Video Solution

Step-by-Step Solution

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{2^4}{2^3}=2^{4-3}=2^1$Remember that any number raised to the 1st power is equal to the number itself, meaning that: