Square Root Quotient Property: Using variables

Express the definition of root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power:

$n=2$

meaning:

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

Proceed to convert the expression using the root definition we mentioned above:

$\frac{\sqrt{25x^2}}{\sqrt{x^2}}=\frac{(25x^2)^{\frac{1}{2}}}{(x^{2)^{\frac{1}{2}}}}$

Now let's recall two laws of exponents:

a. The law of exponents for a power applied to a product inside of parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

b. The law of exponents for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these laws to the numerator and denominator of the fraction in the expression that we obtained in the last step:

$\frac{(25x^2)^{\frac{1}{2}}}{(x^{2)^{\frac{1}{2}}}}=\frac{25^{\frac{1}{2}}\cdot(x^2)^{\frac{1}{2}}}{(x^2)^{\frac{1}{2}}}=\frac{25^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x^{2\cdot\frac{1}{2}}}$

In the first stage we applied the above law of exponents mentioned in a' and then proceeded to apply the power to both factors of the product inside of the parentheses in the fraction's numerator.

We carried this out carefully by using parentheses given that one of the factors in the parentheses is already raised to a power. In the second stage we applied the second law of exponents mentioned in b' to the second factor in the product in the fraction's numerator and similarly to the factor in the fraction's denominator,

Let's simplify all of the expressions that we obtained:

$\frac{25^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x^{2\cdot\frac{1}{2}}}=\frac{\sqrt{25}x^{\frac{2}{2}}}{x^{\frac{2}{2}}}=\frac{5x^1}{x^1}$

In the first stage we converted the fraction's power back to a root. For the first factor in the product, this was done using the definition of root as a power mentioned at the beginning of the solution (in the opposite direction)

We then proceeded to calculate the numerical value of the root.

Additionally - we calculated the product of the power of the second factor in the product in the fraction's numerator in the expression that we obtained. Similarly we carried this out for the factor in the fraction's denominator. We then simplified the resulting fraction for that factor.

Let's complete the calculation and simplify the resulting fraction:

$\frac{5x^1}{x^1} =\frac{5\not{x}}{\not{x}}=5$

Let's summarize the steps of the solution thus far, as seen below:

$\frac{\sqrt{25x^2}}{\sqrt{x^2}}=\frac{(25x^2)^{\frac{1}{2}}}{(x^{2)^{\frac{1}{2}}}} =\frac{5x^1}{x^1} =5$

Therefore the correct answer is answer c.

Express the following root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown

$n=2$

Meaning:

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

Let's return to the problem and use the root definition that we mentioned above to convert the root in the fraction's numerator:

$\frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}$

Remember the two following laws of exponents:

a. The law of exponents for a power applied to a product inside of parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

b. The law of exponents for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these laws to the fraction's numerator in the expression that we obtained in the last step:

$\frac{(49x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}\cdot(x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x}$

$$In the first stage we applied the above-mentioned law of exponents noted in a' and then proceeded to applythe power to both factors of the product (in parentheses) in the fraction's numerator. We we careful to use parentheses given that one of the factors in the parentheses is already raised to a power.

In the second stage we applied the second law of exponents mentioned in b' to the second factor in the product,

Let's simplify the expression that we obtained:

$\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x}=\frac{\sqrt{49}x^{\frac{2}{2}}}{x}=\frac{7x^1}{x}$

In the first stage we converted the fraction's power back to a root, for the first factor in the product, using the definition of root as a power mentioned at the beginning of the solution ( in the opposite direction)

Additionally- we calculated the product in the exponent of the second factor in the product in the fraction's numerator in the expression that we obtained. We then we simplified the resulting fraction.

Finish the calculation and proceed to simplify the resulting fraction:

$\frac{7x^1}{x}=\frac{7\not{x}}{\not{x}}=7$

Let's summarize the various steps of the solution that we obtained thus far, as shown below:

$\frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x} =7$

Therefore the correct answer is answer c.

Express the definition of root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:

$n=2$

Meaning:

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

Let's return to the problem and convert the numerator of the fraction by using the root definition that we mentioned above :

$\frac{\sqrt{x^4}}{x}=\frac{(x^4)^{\frac{1}{2}}}{x}$

Let's recall the power law for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Apply this law to the numerator of the fraction in the expression that we obtained in the last step:

$\frac{(x^4)^{\frac{1}{2}}}{x}=\frac{x^{4\cdot\frac{1}{2}}}{x}=\frac{x^\frac{4}{2}}{x}$

In the first step we applied the above power law and in the second step we performed the multiplication in the power exponent of the numerator term,

Continue to simplify the expression that we obtained. Begin by reducing the fraction with the power exponent in the numerator term and then proceed to apply the power law for division between terms with identical bases:

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

Simplify the fraction in the now complete expression:

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

Let's summarize the various steps of the solution that we obtained: As shown below

$$$\frac{\sqrt{x^4}}{x}=\frac{(x^4)^{\frac{1}{2}}}{x}=\frac{x^2}{x}=x$

Therefore the correct answer is answer A.

To solve the problem, we will simplify the expression step-by-step:

• Step 1: Simplify $\sqrt{36x^4}$
  We know that $\sqrt{36x^4}$ can be rewritten as $\sqrt{36} \cdot \sqrt{x^4}$.
  First, simplify $\sqrt{36}$:
  $\sqrt{36} = 6$ because $6^2 = 36$.
  Next, simplify $\sqrt{x^4}$:
  $\sqrt{x^4} = x^2$ because $(x^2)^2 = x^4$.
• Step 2: Simplify $\sqrt{x^2}$
  $\sqrt{x^2} = x$ (assuming $x \geq 0$ to avoid absolute values).
• Step 3: Simplify the expression $\frac{\sqrt{36x^4}}{\sqrt{x^2}}$
  Substitute the values obtained in the above steps:
  $\frac{\sqrt{36x^4}}{\sqrt{x^2}} = \frac{6x^2}{x}$.
  The expression simplifies to $6x$ as $\frac{x^2}{x} = x$.

Therefore, the solution to the problem is $6x$, which matches choice 4.

Let's solve the problem by following these steps:

• Step 1: Simplify the fraction under the square root.
  The expression is $\frac{100x^4}{25x^2}$. We can simplify this by dealing with the coefficient and the variable separately:
• The numerical part: $\frac{100}{25} = 4$.
• The variable part: $\frac{x^4}{x^2} = x^{4-2} = x^2$, using the laws of exponents.

Thus, the simplified fraction is $4x^2$.

• Step 2: Apply the square root.
  We have $\sqrt{4x^2}$. We apply the square root to both terms:
• $\sqrt{4} = 2$, since 2 squared equals 4.
• $\sqrt{x^2} = x$, assuming $x$ is positive (or using the absolute value to ensure non-negativity, but the context suggests a straightforward approach).

Thus, $\sqrt{4x^2} = 2x$.

• Conclusion:

Therefore, the solution to the expression is $2x$.

To solve this problem, we'll simplify the given expression step by step.

Firstly, observe the expression: $\sqrt{\frac{x^8}{x^4}}$.

• Step 1: Apply the quotient of powers rule: The expression inside the square root is $\frac{x^8}{x^4}$, which simplifies to $x^{8-4} = x^4$ using the rule $\frac{x^m}{x^n} = x^{m-n}$.
• Step 2: Apply the square root rule: Now we have $\sqrt{x^4}$. Utilizing the property of square roots, we find $\sqrt{x^4} = x^{4/2} = x^2$.

Therefore, the simplified expression is $\textbf{x}^2$.

Thus, the final solution to the problem is $\textbf{x}^2$, which corresponds to choice 2 in the given list of options.

To solve this problem, we will follow these steps:

• Step 1: Simplify the fraction under the square root.
• Step 2: Apply the square root property to the simplified result.
• Step 3: Simplify the final expression.

Now, let’s work through each step:

Step 1: Simplify the fraction $\frac{144x^8}{16x^2}$.

Divide the coefficients: $\frac{144}{16} = 9$.

Subtract the exponents of $x$: $x^{8-2} = x^6$.

Thus, the simplified fraction is $9x^6$.

Step 2: Apply the square root property on $9x^6$.

The square root of a product is the product of the square roots, so we have:

$\sqrt{9x^6} = \sqrt{9} \cdot \sqrt{x^6}$.

Step 3: Simplify the result.

$\sqrt{9} = 3$.

$\sqrt{x^6} = x^{\frac{6}{2}} = x^3$.

Therefore, the expression simplifies to $3x^3$.

Thus, the solution to the problem is $3x^3$.

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

• Step 1: Simplify the fraction $\frac{196x^2}{49}$.
• Step 2: Apply the square root quotient property.
• Step 3: Simplify each term to find the final simplified form.

Now, let's perform each step:

Step 1: Simplify the fraction inside the square root:

$\frac{196x^2}{49} = \frac{196}{49} \times x^2$

Simplify $\frac{196}{49}$: Since $196 \div 49 = 4$, we have $\frac{196}{49} = 4$.

The expression therefore simplifies to $4x^2$.

Step 2: Apply the square root quotient property:

$\sqrt{4x^2} = \sqrt{4} \times \sqrt{x^2}$

Step 3: Simplify each term:

$\sqrt{4} = 2$ and $\sqrt{x^2} = x$ (assuming $x \geq 0$).

Therefore, the expression simplifies to $2x$.

Thus, the solution to the problem is $\boxed{2x}$.

Let's solve the problem step by step:

Step 1: Simplify the fraction inside the square root:
$\frac{25x^8}{225x^4}$

Divide both the numerator and the denominator by the greatest common factors. Notice that $25$ and $225$ have a common factor of $25$, and $x^8$ and $x^4$ have a common factor of $x^4$.

The simplification becomes:
$\frac{25 \div 25 \cdot x^{8-4}}{225 \div 25 \cdot x^{4-4}} = \frac{1 \cdot x^4}{9 \cdot 1} = \frac{x^4}{9}$

Step 2: Apply the Quotient Property of Square Roots:
$\sqrt{\frac{x^4}{9}} = \frac{\sqrt{x^4}}{\sqrt{9}}$

Step 3: Simplify the square roots:
Since $\sqrt{x^4} = x^2$ (assuming $x \geq 0$ as square roots imply non-negative results), and $\sqrt{9} = 3$,
$\frac{\sqrt{x^4}}{\sqrt{9}} = \frac{x^2}{3}$

Therefore, the solution to the problem is $\frac{1}{3}x^2$.

To solve this problem, we'll divide this task into clear steps:

• Step 1: Rewrite the expression using the square root property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. So, $\sqrt{\frac{225x^4}{9x^2}} = \frac{\sqrt{225x^4}}{\sqrt{9x^2}}$.
• Step 2: Simplify $\sqrt{225x^4}$. This can be broken down as $\sqrt{225} \times \sqrt{x^4}$.
• Step 3: Simplify each part: $\sqrt{225} = 15$ and $\sqrt{x^4} = x^2$ since $(x^2)^2 = x^4$.
• Step 4: Now, simplify $\sqrt{9x^2}$. This is $\sqrt{9} \times \sqrt{x^2}$.
• Step 5: Simplify these parts: $\sqrt{9} = 3$ and $\sqrt{x^2} = x$ (assuming $x > 0$ for simplification purposes).
• Step 6: Putting it all together, $\frac{\sqrt{225x^4}}{\sqrt{9x^2}} = \frac{15x^2}{3x}$.
• Step 7: Simplify the fraction: $\frac{15x^2}{3x} = 5x$ (since $\frac{x^2}{x} = x$).

Therefore, the solution to the problem is $5x$.

In the choices provided, the correct answer that matches our solution is choice 2: $5x$.

To solve this problem, we will proceed as follows:

• Step 1: Simplify the expression inside the square root.
• Step 2: Apply the square root to both the numerator and the denominator separately.
• Step 3: Simplify the resulting expression.

Let's go through each step:

Step 1: Simplify the fraction $\frac{144x^{10}}{9x^4}$.

In the given expression $\frac{144x^{10}}{9x^4}$, separate the numeric part from the variable part:

$\frac{144}{9} = 16$ and $\frac{x^{10}}{x^4} = x^{10-4} = x^6$.

Thus, the expression simplifies to $16x^6$.

Step 2: Apply the square root property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to each part.

$\sqrt{\frac{144x^{10}}{9x^4}} = \sqrt{16x^6}$.

Now separate into numeric and variable components: $\sqrt{16}\cdot\sqrt{x^6}$.

$\sqrt{16} = 4$ because $4^2 = 16$.

$\sqrt{x^6} = (x^6)^{1/2} = x^{6/2} = x^3$.

Step 3: Combine the results.

Combine the simplified components: $4x^3$.

Therefore, the solution to the problem is $\boxed{4x^3}$.

To solve this problem, we'll simplify the expression inside the square root step-by-step:

• Step 1: Simplify the fraction $\frac{64x^2}{x^4}$ using the quotient rule for exponents:
• $\frac{64x^2}{x^4} = 64 \cdot x^{2-4} = 64 \cdot x^{-2}$.
• Step 2: Now apply the square root to the simplified expression:
• $\sqrt{\frac{64x^2}{x^4}} = \sqrt{64x^{-2}} = \sqrt{64} \cdot \sqrt{x^{-2}}$.
• $\sqrt{64} = 8$ and $\sqrt{x^{-2}} = x^{-1}$ (since it represents inverse squaring).
• Thus, $\sqrt{64x^{-2}} = 8 \cdot x^{-1} = \frac{8}{x}$.

Therefore, the solution to the problem is $\frac{8}{x}$, which matches choice 1.

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

• Step 1: Simplify the expression inside the square root.
• Step 2: Apply the square root property to the simplified expression.
• Step 3: Confirm the result with the answer choices.

Now, let's work through each step:

Step 1: Simplify the expression inside the square root:
We have $\frac{64x^4}{16x^2}$. Divide the coefficients and the powers of $x$:
$\frac{64}{16} = 4$ and using exponents, $\frac{x^4}{x^2} = x^{4-2} = x^2$.
Therefore, $\frac{64x^4}{16x^2} = 4x^2$.

Step 2: Apply the square root property:
$\sqrt{4x^2} = \sqrt{4} \cdot \sqrt{x^2} = 2 \cdot x = 2x$.

Therefore, the solution to the problem is $2x$.

To solve this mathematical expression, follow these steps:

• Step 1: Use the Quotient Property of Exponents

  Simplify the expression inside the square root using the rule:
  $\frac{x^8}{x^{10}} = x^{8-10} = x^{-2}$.

• Step 2: Apply the Square Root Property

  Now, apply the square root:
  $\sqrt{x^{-2}} = (x^{-2})^{1/2} = x^{-2 \cdot \frac{1}{2}} = x^{-1}$.

• Step 3: Express in Simpler Form

  The expression $x^{-1}$ can be written as $\frac{1}{x}$.

Therefore, the final simplified form of the expression is $\frac{1}{x}$.

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

• Simplify the expression inside the square root using exponent rules.
• Evaluate the square root of the simplified expression.

Let's work through the solution step-by-step:

First, simplify the expression inside the square root. We have:

$\frac{x^{20}}{x^{24}} = x^{20-24} = x^{-4}$

Next, apply the square root to the simplified expression:

$\sqrt{x^{-4}} = \left(x^{-4}\right)^{1/2} = x^{-4 \times \frac{1}{2}} = x^{-2}$

This can be written as:

$x^{-2} = \frac{1}{x^2}$

Therefore, the solution to the problem is $\frac{1}{x^2}$.