Power of a Quotient Rule for Exponents - Examples, Exercises and Solutions

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:

$b^1=b$

Therefore, in the problem we obtain:

$2^1=2$

Therefore, the correct answer is option a.

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$

Let's apply it to the problem:

$\frac{9^9}{9^3}=9^{9-3}=9^6$$$

Therefore, the correct answer is b.

To solve the expression $\frac{13^{17}}{13^{14}}$, we use the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a non-zero number, and $m$ and $n$ are integers.

In the given expression, $a = 13$, $m = 17$, and $n = 14$. Applying the power of a quotient rule, we perform the following calculation:

Subtract the exponent in the denominator from the exponent in the numerator: $17 - 14 = 3$.

This simplification leads us to:

$13^{17-14} = 13^3$

Therefore, the final simplified expression is $13^3$.

Since a division operation between two terms with identical bases is required, we will use the power property to divide terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$$\frac{c^m}{c^n}=c^{m-n}$Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the power property:

$\frac{a^4}{a^{-6}}=a^{4-(-6)}=a^{4+6}=a^{10}$Therefore, the correct answer is option C.

The given expression is $\frac{8^{16}}{8^8}$. To solve this, we apply the Power of a Quotient Rule for Exponents.

This rule states that when dividing two exponential expressions with the same base, we subtract the exponent of the denominator from the exponent of the numerator. Mathematically, it can be expressed as:

• $\frac{a^m}{a^n} = a^{m-n}$

In this problem, the base $8$ is the same in both the numerator and the denominator, so we can apply this rule.

Subtract the exponent of the denominator from the exponent of the numerator:

• $16 - 8 = 8$

Therefore, the simplified form of the given expression is:

• $8^8$

Thus, the answer is $8^8$.