Rules of Roots Combined: Solving the equation

Introduction:

We will address the following two laws of exponents:

a. Definition of root as an exponent:

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

b. The law of exponents for an exponent applied to terms in parentheses:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

Note:

By combining the two laws of exponents mentioned in a (in the first and third stages below) and b (in the second stage below), we can derive another new rule:

$\sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}$

Specifically for the fourth root we obtain the following:

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

Therefore, we can proceed to solve the problem:

$\frac{\sqrt{64}}{\sqrt{4}}=2x$

Let's start by simplifying the expression on the left side, using the new rule that we studied in the introduction:

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

( However this time in the opposite direction, meaning we'll insert the product of roots as a product of terms under the same root) Then we'll perform the multiplication under the root:

$\frac{\sqrt{64}}{\sqrt{4}}=2x \\ \sqrt{\frac{64}{4}}=2x \\ \sqrt{16}=2x \\ 4=2x \\$In the final stage, we used the known fourth root of the number 16,

After simplifying the expression on the left side, to isolate the unknown, we'll divide both sides of the equation by its coefficient:

$4=2x\hspace{6pt}\text{/}:2 \\ 2=x \\ \downarrow\\ \boxed{x=2}$

Let's summarize the solution of the equation:

$\frac{\sqrt{64}}{\sqrt{4}}=2x \\ \sqrt{16}=2x \\ 4=2x \\ \downarrow\\ \boxed{x=2}$$$

Therefore, the correct answer is answer b.

To solve this problem, let's proceed with the following steps:

• Step 1: Start with the given equation:
  $\frac{\sqrt{98}}{\sqrt{x}} = 7$.
• Step 2: Apply the square root property to combine the fraction:
  $\sqrt{\frac{98}{x}} = 7$.
• Step 3: Square both sides to eliminate the square root:
  $\frac{98}{x} = 49$.
• Step 4: Solve for $x$ by multiplying both sides by $x$:
  $98 = 49x$.
• Step 5: Isolate $x$ by dividing both sides by 49:
  $x = \frac{98}{49}$.
• Step 6: Simplify the fraction:
  $x = \frac{98}{49} = 2$.

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

To solve the equation $\sqrt{6}x = \sqrt{36}$, we will proceed with the following steps:

• Step 1: Simplify the square root on the right-hand side.
  $\sqrt{36} = 6$.
• Step 2: Substitute the simplified value back into the equation to obtain:
  $\sqrt{6}x = 6$.
• Step 3: Solve for $x$ by isolating the variable. Divide both sides by $\sqrt{6}$:
  $x = \frac{6}{\sqrt{6}}$.
• Step 4: Simplify the fraction:
  Multiply the numerator and denominator by $\sqrt{6}$:
  $x = \frac{6 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}} = \frac{6 \sqrt{6}}{6} = \sqrt{6}$.

Therefore, the solution to the equation is $x = \sqrt{6}$.

To solve this problem, we'll apply the properties of square roots and some straightforward algebraic techniques:

Step 1: Recall the equation given is:

$\frac{\sqrt{90}}{\sqrt{x}} = 3$

Use the property of the square root quotient:

$\sqrt{\frac{90}{x}} = 3$

Step 2: To eliminate the square root, square both sides of the equation:

$\left(\sqrt{\frac{90}{x}}\right)^2 = 3^2$

Thus, we have:

$\frac{90}{x} = 9$

Step 3: Solve for $x$ by performing algebraic manipulation:

Multiply both sides by $x$ to remove the fraction:

$90 = 9x$

Divide both sides by 9 to isolate $x$:

$x = \frac{90}{9}$

Simplifying, we find:

$x = 10$

Therefore, the solution to the equation is $x = 10$.

To solve the given equation, $\frac{\sqrt{50}}{\sqrt{x}} = 5$, we will follow these steps:

• Step 1: Multiply both sides by $\sqrt{x}$ to isolate $\sqrt{x}$:
  $\sqrt{50} = 5\sqrt{x}$.
• Step 2: Divide both sides by 5 to solve for $\sqrt{x}$:
  $\sqrt{x} = \frac{\sqrt{50}}{5}$.
• Step 3: Simplify $\frac{\sqrt{50}}{5}$:
  $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}$. Thus, $\frac{5\sqrt{2}}{5} = \sqrt{2}$.
• Step 4: Square both sides to solve for $x$:
  $x = (\sqrt{2})^2 = 2$.

Now, compare with the provided choices:
Choice b is $\sqrt{4}$, which simplifies to 2.
Choice c is 2. Both represent the correct answer.

Therefore, the correct answer is Answers b and c.

Examine the following two laws of exponents:

a. Definition of root as an exponent:

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

b. The law of exponents for an exponent applied to terms in parentheses:

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

Note:

By combining these two laws of exponents mentioned in a (in the first and third steps ahead) and b (in the second step ahead), we can obtain a new rule:

$\sqrt[n]{a\cdot b}=\\ (a\cdot b)^{\frac{1}{n}}=\\ a^{\frac{1}{n}}\cdot b^{\frac{1}{n}}=\\ \sqrt[n]{a}\cdot \sqrt[n]{ b}\\ \downarrow\\ \boxed{\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{ b}}$

And specifically for the fourth root we obtain the following:

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$$$

$$Therefore, we will proceed with solving the problem as follows:

$\frac{x}{\sqrt{6}} = \sqrt{2}\cdot\sqrt{3}$$$

First, we'll eliminate the fraction line, which we'll achieve by multiplying both sides of the equation by the common denominator which is- $\sqrt{6}$:

$\frac{x}{\sqrt{6}} = \sqrt{2}\cdot\sqrt{3} \hspace{6pt}\text{/}\cdot\sqrt{6}\\ x=\sqrt{2}\cdot\sqrt{3}\cdot \sqrt{6}$

Let's continue to simplify the expression on the left side of the equation, using the following rule :

(which of course also applies to multiplication between numbers under a root), next we'll perform the multiplication under the root:

$x=\sqrt{2}\cdot\sqrt{3}\cdot\sqrt{6} \\ x=\sqrt{2\cdot3\cdot6} \\ x=\sqrt{36}\\ \boxed{x=6}$

In the final step, we used the known fourth root of the number 36,

Let's summarize the solution of the equation:

$\frac{x}{\sqrt{6}} = \sqrt{2}\cdot\sqrt{3} \hspace{6pt}\text{/}\cdot\sqrt{6}\\ x=\sqrt{2}\cdot\sqrt{3}\cdot \sqrt{6} \\ x=\sqrt{36}\\ \boxed{x=6}$

$$Therefore, the correct answer is answer a.

To solve the equation \<$\frac{\sqrt{20}\cdot\sqrt{5}}{x} = 2\cdot\sqrt{25}$\>, follow these steps:

• Step 1: Simplify the left-hand side.
  - Use the product rule for roots: $\sqrt{20} \cdot \sqrt{5} = \sqrt{20 \cdot 5} = \sqrt{100}$.
• Step 2: Simplify $\sqrt{100}$ to get 10\.
• Step 3: Substitute and simplify the equation:
  \(\frac{10}{x} = 2 \cdot \sqrt{25}.
• Step 4: Simplify the right-hand side:
  $\sqrt{25} = 5$ so $2 \cdot 5 = 10$.
• Step 5: Equate both sides:
  $\frac{10}{x} = 10$.
• Step 6: Solve for $x$:
  Multiply both sides by $x$, then divide by 10:
  $10 = 10x$ produces $x = 1$.

Therefore, the solution to the problem is $\boldsymbol{x = 1}$.

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

• Simplify the expression $\sqrt{8}\cdot\sqrt{4}\cdot\sqrt{2}$.
• Divide this product by $\sqrt{2}$.
• Set the result equal to $\sqrt{x^2}$ and solve for $x$.

Now, let's work through each step:

Step 1: First, simplify the product under the square root:

$\sqrt{8}\cdot\sqrt{4}\cdot\sqrt{2} = \sqrt{8\cdot4\cdot2}$.

This simplifies to:

$\sqrt{64}$, because $8 \cdot 4 \cdot 2 = 64$.

Step 2: Now, divide by $\sqrt{2}$:

$\frac{\sqrt{64}}{\sqrt{2}} = \sqrt{\frac{64}{2}} = \sqrt{32}$.

Step 3: Equate this to $\sqrt{x^2}$:

$\sqrt{32} = \sqrt{x^2}$.

This implies $|x| = \sqrt{32}$, giving us two possible solutions: $x = \sqrt{32}$ and $x = -\sqrt{32}$.

Since the simplification naturally leads to positive expressions, we find:

The solution is $x = \sqrt{32}$.

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

• Step 1: Simplify the left side of the equation, $\sqrt{\sqrt{81}}$.
• Step 2: Simplify the right side of the equation, $\sqrt[3]{\sqrt{x^6}}$.
• Step 3: Equate the simplified expressions and solve for $x$.

Now, let's simplify each side:

Step 1: Simplify $\sqrt{\sqrt{81}}$.

First, evaluate $\sqrt{81}$, which is $9$, since $9^2 = 81$.
Then, evaluate $\sqrt{9}$, which is $3$, since $3^2 = 9$.
So, $\sqrt{\sqrt{81}} = 3$.

Step 2: Simplify $\sqrt[3]{\sqrt{x^6}}$.

Express $\sqrt{x^6}$ as $(x^6)^{1/2} = x^{6/2} = x^3$.
Express $\sqrt[3]{x^3}$ as $(x^3)^{1/3} = x^{3/3} = x^1 = x$.
So, $\sqrt[3]{\sqrt{x^6}} = x$.

Step 3: Set the simplified expressions equal.

We have simplified both sides of the equation to get $3 = x$.
Therefore, the solution to the problem is $x = 3$.

Hence, the correct answer is $x = 3$.

Therefore, the correct choice is:

Choice 2: $x = 3$.

Introduction:

We will address the following three laws of exponents:

a. Definition of root as an exponent:

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

b. The law of exponents for an exponent applied to a product in parentheses:

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

c. The law of exponents for an exponent applied to a quotient in parentheses:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

Note:

(1). By combining the two laws of exponents mentioned in a (in the first and third steps ) and b (in the second step ), we can obtain a new rule:

$\sqrt[n]{a\cdot b}=\\ (a\cdot b)^{\frac{1}{n}}=\\ a^{\frac{1}{n}}\cdot b^{\frac{1}{n}}=\\ \sqrt[n]{a}\cdot \sqrt[n]{ b}\\ \downarrow\\ \boxed{\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{ b}}$

Specifically for the fourth root we obtain the following::

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$

(2). Similarly, note that by combining the two laws of exponents mentioned in a (in the first and third steps later) and c (in the second step later), we can obtain another new rule:

$\sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}$

Specifically for the fourth root we obtain the following:

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

Therefore, in solving the problem, meaning - in simplifying the given expression, we will apply the two new rules that we received in the introduction:

(1).

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$

(2).

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

We will start by simplifying the expression in the numerator using the rule that we examined in the introduction (1) (however this time in the opposite direction, meaning we will insert the product of roots as a product of terms under the same root) Then we will proceed to perform the multiplication under the root in the numerator:

$\frac{\sqrt{4}\cdot\sqrt{5}}{\sqrt{10}}= \\ \frac{\sqrt{4\cdot5}}{\sqrt{10}}= \\ \frac{\sqrt{20}}{\sqrt{10}}= \\$We will then simplify the fraction, using the second rule that we examined in the introduction (2) (once again in the opposite direction, meaning we will insert the quotient of roots as a quotient of terms under the same root) Then we will proceed to reduce the fraction under the root:

$\frac{\sqrt{20}}{\sqrt{10}}= \\ \sqrt{\frac{20}{10}}=\\ \boxed{\sqrt{2}}$

Summarize the process of simplifying the expression in the problem:

$\frac{\sqrt{4}\cdot\sqrt{5}}{\sqrt{10}}= \\ \frac{\sqrt{20}}{\sqrt{10}}= \\ \boxed{\sqrt{2}}$

Therefore, the correct answer is answer c.

To solve $\frac{\sqrt{4} \cdot \sqrt{5}}{\sqrt{25}} = 2x$, we follow these steps:

• Simplify $\sqrt{4}$: The square root of 4 is 2.
• Simplify $\sqrt{5}$: $\sqrt{5}$ remains $\sqrt{5}$.
• Simplify $\sqrt{25}$: The square root of 25 is 5.
• Substitute these values back: $\frac{2 \cdot \sqrt{5}}{5} = 2x$.
• Write the expression: $\frac{2\sqrt{5}}{5} = 2x$.
• Divide both sides by 2 to solve for $x$:
$x = \frac{2\sqrt{5}}{5 \cdot 2} = \frac{\sqrt{5}}{5}$

The simplified expression for $\sqrt{5}$ is equivalent to $\frac{\sqrt{20}}{10}$, using $\sqrt{5} = \frac{\sqrt{20}}{2}$ since $\sqrt{20} = 2\sqrt{5}$.

Therefore, the solution to the problem is $\boxed{\frac{\sqrt{20}}{10}}$.

Let's solve the equation step by step:

• Step 1: Simplify the left side of the equation
  We are given $\frac{\sqrt{2}\cdot\sqrt{4}}{\sqrt{16}}$. Start by simplifying each square root:
  • $\sqrt{2}$ remains as it is.
  • $\sqrt{4} = 2$ because $4 = 2^2$.
  • $\sqrt{16} = 4$ because $16 = 4^2$.
  Substitute these into the initial expression: $\frac{\sqrt{2} \cdot 2}{4} = \frac{2\sqrt{2}}{4}$.
• Step 2: Simplify the expression
  The expression $\frac{2\sqrt{2}}{4}$ simplifies to $\frac{\sqrt{2}}{2}$.
• Step 3: Equate to the right side
  Set this equal to the right side: $\frac{\sqrt{2}}{2} = \frac{x}{\sqrt{8}}$.
• Step 4: Cross-multiply to solve for $x$
  Cross-multiplying gives $x \cdot 2 = \sqrt{2} \cdot \sqrt{8}$. Simplify the right side:
  • $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$
  • So, $\sqrt{2} \cdot \sqrt{8} = \sqrt{2} \cdot 2\sqrt{2} = 2 \cdot (\sqrt{2} \cdot \sqrt{2}) = 2 \cdot 2 = 4$
  • The equation becomes $2x = 4$.
  • Solving for $x$, divide both sides by 2: $x = \frac{4}{2} = 2$.

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

To solve the problem $\sqrt{\frac{16}{\sqrt[3]{64}}} = \sqrt{x^2}$, we proceed step-by-step as follows:

• First, we simplify $\sqrt[3]{64}$. Since $64 = 4^3$, it follows that $\sqrt[3]{64} = 4$.
• Next, simplify the expression $\frac{16}{\sqrt[3]{64}}$:
  $\frac{16}{4} = 4$.
• Now, take the square root of this simplified value. Thus, $\sqrt{4} = 2$.
• The equation simplifies to: $\sqrt{x^2} = 2$. Since $\sqrt{x^2} = |x|$, we have $|x| = 2$.
• This implies $x = 2$ or $x = -2$.
• However, choices include only positive solutions, and thus $x = 2$.

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

To solve this problem, we'll begin by simplifying both sides of the equation:

• Simplifying the left-hand side:

$\sqrt[5]{\sqrt{x^{10}}}$ can be rewritten using properties of exponents and roots. The inner square root is $(x^{10})^{1/2} = x^{10 \cdot \frac{1}{2}} = x^{5}$.

Then, take the fifth root: $(x^{5})^{\frac{1}{5}} = x^{5 \cdot \frac{1}{5}} = x^{1} = x$.
Thus, the left-hand side simplifies to $x$.

• Simplifying the right-hand side:

$\sqrt{\sqrt{81}}$ simplifies as follows: First, find $\sqrt{81} = 9$.
Then, compute $\sqrt{9} = 3$.

So, the equation reduces to $x = 3$.

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

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

• Step 1: Simplify $\sqrt{\sqrt{16}}$.
• Step 2: Simplify $\sqrt{25}$.
• Step 3: Calculate the right-hand side.
• Step 4: Solve $\sqrt{x^6} =$ computed right-hand side.

Now, let's work through each step:
Step 1: Simplify $\sqrt{\sqrt{16}}$.
We know that $\sqrt{16} = 4$, and $\sqrt{4} = 2$. So, $\sqrt{\sqrt{16}} = 2$.

Step 2: Simplify $\sqrt{25}$.
We know that $\sqrt{25} = 5$.

Step 3: Calculate the entire right-hand side.
We have $\sqrt{\sqrt{16}} \cdot \sqrt{25} = 2 \cdot 5 = 10$.

Step 4: Solve $\sqrt{x^6} = 10$.
Rewrite the left side as $(x^6)^{1/2} = x^{6/2} = x^3$.
Thus, $x^3 = 10$.

Therefore, the solution to the problem is $x^3 = 10$.

To solve this equation, we will follow these steps:

• Step 1: Simplify the left side of the equation $\sqrt{144}$.
• Step 2: Simplify the right side of the equation $\sqrt[3]{\sqrt[5]{x^{30}}}$.
• Step 3: Equate the simplified expressions and solve for $x$.

Let us go through these steps:

Step 1: Simplify the left side:

The left side of the equation is $\sqrt{144}$, which simplifies to $12$, because $\sqrt{144} = 12$.

Step 2: Simplify the right side:

The expression on the right is $\sqrt[3]{\sqrt[5]{x^{30}}}$. Let's simplify it step by step:

• First, simplify $\sqrt[5]{x^{30}}$:
  - Using the rule $\sqrt[n]{a^m} = a^{m/n}$, we have $\sqrt[5]{x^{30}} = x^{30/5} = x^6$.
• Next, simplify $\sqrt[3]{x^6}$:
  - Again using the same rule, $\sqrt[3]{x^6} = x^{6/3} = x^2$.

Step 3: Equate and solve:

From the previous steps, we get:

$12 = x^2$

Therefore, the solution to the equation is:
$x^2 = 12$.