Power of Quotient Rule Practice Problems & Division of Exponents

We need to simplify the expression $\frac{5^3}{5^8}$ using the rules of exponents. Specifically, we will use the power of a quotient rule for exponents which states that when you divide like bases you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$.

Here, the base is 5, the exponent in the numerator is 3, and the exponent in the denominator is 8.

• Apply the rule: $5^{3-8}$
• Subtract the exponents: $5^{-5}$.

Therefore, the simplified expression is $5^{-5}$.

The solution to the question is: $5^{-5}$

To solve the expression $\frac{5^7}{5^{10}}$, we need to apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases, you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$.

In this particular case, the base is 5, and the exponents are 7 and 10. Using the rule, we subtract the exponent in the denominator from the exponent in the numerator:

• Numerator exponent = 7
• Denominator exponent = 10

The solution to the question is: $5^{7-10}$.

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 $13^3$.

To solve the expression $\frac{17^{16}}{17^{20}}$, we can apply the Power of a Quotient Rule for Exponents. This rule states that when you divide two exponents with the same base, you can subtract the exponents to simplify the expression.

The given expression is:

$\frac{17^{16}}{17^{20}}$

According to the Quotient Rule for Exponents, this expression can be simplified as:

$17^{16-20}$

Here's the step-by-step breakdown:

• The base of both the numerator and the denominator is the same, that is, 17.
• According to the rule, subtract the exponent in the denominator from the exponent in the numerator: $16 - 20$.
• This gives us the exponent: $-4$.

So, the simplified expression is:

$17^{-4}$

However, as requested, we should express this as:

$17^{16-20}$

The solution to the question is:

$17^{16-20}$

To solve the expression $\frac{25^9}{25^2}$$$, we will use the Power of a Quotient Rule for Exponents. According to this rule, when dividing like bases, we subtract the exponents.

• $a^m \div a^n = a^{m-n}$

In the given expression, the base $25$ is the same for both the numerator and the denominator. Therefore, we can apply the rule as follows:

• Identify the exponents: $m = 9$ and $n = 2$.

• Subtract the exponents: $9 - 2 = 7$.

• Write the result as a single power of the base: $25^7$.

Thus, the expression $\frac{25^9}{25^2}$ simplifies to $25^7$.

The solution to the question is: 25^7