Square Root Quotient Property: Same base and different indicator

Let's simplify the expression $\frac{\sqrt{5}}{\sqrt[4]{5}}$ using the rules of exponents:

• First, we convert $\sqrt{5}$ to exponent form: $\sqrt{5} = 5^{1/2}$.
• Next, we convert $\sqrt[4]{5}$ to exponent form: $\sqrt[4]{5} = 5^{1/4}$.
• Now, divide the two expressions: $\frac{5^{1/2}}{5^{1/4}} = 5^{1/2 - 1/4}$.
• Subtract the exponents: $5^{1/2 - 1/4} = 5^{2/4 - 1/4} = 5^{1/4}$.

The problem is simplified to $\sqrt[4]{5}$.

Therefore, the simplified form of the given expression is $\sqrt[4]{5}$.

Express the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Apply this definition and proceed to convert the roots in the problem:

$\frac{\sqrt[4]{2}}{\sqrt[5]{2}}= \frac{2^{\frac{1}{4}}}{2^{\frac{1}{5}}}$

Below is the law of powers for division with identical bases:

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

Let's apply this law to our problem:

$\frac{2^{\frac{1}{4}}}{2^{\frac{1}{5}}}=2^{\frac{1}{4}-\frac{1}{5}}$

In order to not overly complicate our calculations proceed to solve the expression in the power numerator from the last step separately and calculate the value of the fraction:

$\frac{1}{4}-\frac{1}{5}=\frac{5\cdot1-4\cdot1}{20}=\\ \frac{5-4}{20}=\frac{1}{20}$

In the first step, we combined the two fractions into one fraction line, by expanding to the common denominator of 20 and performing the subtraction operation. (In the first fraction on the left we expanded both the numerator and denominator by 5, and in the second fraction we expanded both the numerator and denominator by 4) We then proceeded to simplify the resulting expression,

Returning once more to our problem, consider the result of the subtraction operation between the fractions that we just performed, as shown below:

$2^{\frac{1}{4}-\frac{1}{5}}=2^{\frac{1}{20}}$

Summarize the various steps of the solution:

$\frac{\sqrt[4]{2}}{\sqrt[5]{2}}=2^{\frac{1}{4}-\frac{1}{5}}=2^{\frac{1}{20}}$

Therefore, the correct answer is answer C.

To solve the given problem, let us proceed step by step:

• Step 1: Convert the given roots into fractional exponents:
  The cube root $\sqrt[3]{5}$ can be expressed as $5^{1/3}$.
  The fourth root $\sqrt[4]{5}$ can be expressed as $5^{1/4}$.
• Step 2: Apply the quotient rule for exponents:
  For division, $\frac{a^m}{a^n} = a^{m-n}$.
  Thus, $\frac{5^{1/3}}{5^{1/4}} = 5^{1/3 - 1/4}$.
• Step 3: Perform the subtraction of the exponents:
  $1/3 - 1/4 = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$.
  Therefore, we have $5^{\frac{1}{12}}$.
• Step 4: Compare with answer choices:
  - The expression $5^{\frac{1}{12}}$ directly matches Choice 3.
  - Alternatively, we can express this as a 12th root: $\sqrt[12]{5}$, which matches Choice 1.

Both answer choices (a) $\sqrt[12]{5}$ and (c) $5^{\frac{1}{12}}$ correctly represent the simplified form of the expression.

Thus, the correct solution to the problem is given by Answers (a) and (c).

To solve this problem, we'll follow these steps:

• Step 1: Convert the given roots into fractional exponents.

• Step 2: Simplify the fractional exponents where necessary.

• Step 3: Divide using the properties of exponents.

• Step 4: Simplify the resulting expression.

Now, let's work through each step:

Step 1: Convert the roots to fractional exponents.
The expression $\frac{\sqrt[6]{5^3}}{\sqrt[3]{5^3}}$ becomes $\frac{(5^3)^{\frac{1}{6}}}{(5^3)^{\frac{1}{3}}}$.

Step 2: Simplify each fractional exponent.
We know $(a^m)^n = a^{m \cdot n}$. So, apply this rule: -

$(5^3)^{\frac{1}{6}} = 5^{3 \cdot \frac{1}{6}} = 5^{\frac{3}{6}} = 5^{\frac{1}{2}}$.

$(5^3)^{\frac{1}{3}} = 5^{3 \cdot \frac{1}{3}} = 5^{\frac{3}{3}} = 5^1 = 5$.

Step 3: Divide using the properties of exponents.
The division of powers with the same base: $a^m / a^n = a^{m-n}$. Thus, $\frac{5^{\frac{1}{2}}}{5}$ simplifies to $5^{\frac{1}{2} - 1}$.

Step 4: Simplify the resulting exponent expression.
$5^{\frac{1}{2} - 1} = 5^{-\frac{1}{2}} = \frac{1}{5^{\frac{1}{2}}} = \sqrt{5}$, (since the negative exponent indicates reciprocal).

Therefore, the solution to the problem is $\sqrt{5}$, which matches Choice 2.

To solve this problem, we'll follow these steps:

• Step 1: Convert the given radical expressions into their equivalent fractional power forms.
• Step 2: Use the properties of exponents to simplify the expression.
• Step 3: Convert the result back into a radical form if necessary.

Let's execute these steps:
Step 1: Write $\sqrt[3]{6}$ and $\sqrt{6}$ in terms of fractional powers:
$\sqrt[3]{6} = 6^{1/3}$ and $\sqrt{6} = 6^{1/2}$.

Step 2: Apply the quotient rule for exponents:
$\frac{6^{1/3}}{6^{1/2}} = 6^{1/3 - 1/2} = 6^{\frac{1}{3} - \frac{1}{2}} = 6^{\frac{2 - 3}{6}} = 6^{-\frac{1}{6}}$.

Step 3: Simplify and express back in radical form:
Since a negative exponent denotes the reciprocal, we have $6^{-\frac{1}{6}}$ = $\frac{1}{6^{\frac{1}{6}}}$, which simplifies to $\sqrt[6]{6}$.

Therefore, the expression simplifies to $\sqrt[6]{6}$.

Thus, the solution to the problem is $\sqrt[6]{6}$, which corresponds to choice 3.

To solve this problem, we'll follow these steps:

• Step 1: Convert the roots to exponent notation
• Step 2: Apply the quotient of powers rule
• Step 3: Simplify the expression

Now, let's work through each step:
Step 1: Convert the roots to exponent notation:
$\sqrt{4} = 4^{1/2}$ and $\sqrt[3]{4} = 4^{1/3}$.
Step 2: Calculate the quotient using the rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{4^{1/2}}{4^{1/3}} = 4^{(1/2) - (1/3)}$

Step 3: Simplify the exponent:

$1/2 - 1/3 = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$

Therefore, the expression simplifies to:

$4^{\frac{1}{6}}$

The correct answer is choice 1: $4^{\frac{1}{6}}$.

To solve this problem, we'll follow these steps:

• Step 1: Convert the roots to fractional exponents.
• Step 2: Simplify the fractional exponents using properties of exponents.
• Step 3: Determine the simplified expression and verify the answer choice.

Now, let's work through each step:

Step 1: Convert the roots to fractional exponents.
The cube root $\sqrt[3]{6^2}$ can be expressed as $(6^2)^{1/3} = 6^{2/3}$.

The fourth root $\sqrt[4]{6^2}$ can be written as $(6^2)^{1/4} = 6^{2/4} = 6^{1/2}$.

Step 2: Simplify the quotient of these fractional exponents.
We have $\frac{6^{2/3}}{6^{1/2}}$.

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we get:

$6^{2/3 - 1/2}$.

Step 3: Calculate $2/3 - 1/2$.
To subtract these fractions, find a common denominator. The common denominator of 3 and 2 is 6.

- Convert $2/3$ to sixths: $\frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}$

- Convert $1/2$ to sixths: $\frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$

Perform the subtraction: $\frac{4}{6} - \frac{3}{6} = \frac{1}{6}$.

The expression simplifies to $6^{1/6}$.

Therefore, the simplified answer is $\mathbf{6^{\frac{1}{6}}}$, which corresponds to the correct choice: 3.

The solution to the problem is $\mathbf{6^{\frac{1}{6}}}$.

To solve this problem, we'll transform the given roots into expressions with fractional exponents and then simplify using the rules of exponents.

• Step 1: Express roots as fractional exponents - $\sqrt[5]{36} = 36^{1/5}$ - $\sqrt[10]{36} = 36^{1/10}$

• Step 2: Apply the quotient rule for exponents - We simplify $\frac{36^{1/5}}{36^{1/10}}$ using the property: $\frac{a^m}{a^n} = a^{m-n}$: $\frac{36^{1/5}}{36^{1/10}} = 36^{1/5 - 1/10}$

• Step 3: Simplify the exponent - First, find a common denominator for the exponents: $1/5 = 2/10$ - The subtraction gives us: $2/10 - 1/10 = 1/10$

Thus, the simplified expression is $36^{1/10}$.

Therefore, the solution to the problem is $36^{\frac{1}{10}}$.

To solve this problem, we must simplify the expression $\frac{\sqrt[4]{3}}{\sqrt[6]{3}}$. We will follow these steps:

• Step 1: Convert roots to exponents.
  $\sqrt[4]{3}$ is equivalent to $3^{\frac{1}{4}}$, and $\sqrt[6]{3}$ is equivalent to $3^{\frac{1}{6}}$.
• Step 2: Apply the quotient of powers formula.
  Using the property $\frac{a^m}{a^n} = a^{m-n}$, we have: $\frac{3^{\frac{1}{4}}}{3^{\frac{1}{6}}} = 3^{\frac{1}{4} - \frac{1}{6}}$
• Step 3: Perform the subtraction of the exponents.
  To subtract the fractions $\frac{1}{4}$ and $\frac{1}{6}$, find a common denominator. The least common multiple of 4 and 6 is 12, so: $\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$ Thus, $\frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}$
• Step 4: Simplify the result.
  Thus, we have: $3^{\frac{1}{4} - \frac{1}{6}} = 3^{\frac{1}{12}}$

Hence, the simplified form of the given expression is $3^{\frac{1}{12}}$.

To solve this problem, we will convert the roots into exponent form and simplify:

• Step 1: Express each root as an exponent.

• Step 2: Simplify the expression using the properties of exponents.

Let's apply each step:

Step 1: Convert $\sqrt[3]{4}$ and $\sqrt[6]{4}$ into exponential form:

$\sqrt[3]{4} = 4^{\frac{1}{3}} \quad \text{and} \quad \sqrt[6]{4} = 4^{\frac{1}{6}}$

Step 2: Apply the quotient rule for exponents $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{4^{\frac{1}{3}}}{4^{\frac{1}{6}}} = 4^{\frac{1}{3} - \frac{1}{6}}$

We need to subtract the exponents. First, find a common denominator for the fractions:

$\frac{1}{3} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} = \frac{1}{6}$

The expression simplifies to:

$4^{\frac{1}{6}}$

Therefore, the simplified result is $4^\frac{1}{6}$.

Comparing this with the answer choices, we conclude that the correct choice is $4^{\frac{1}{6}}$.