Rules of Roots Combined - Examples, Exercises and Solutions

Let's begin by calculating the numerical value of each of the roots in the given options:

$\sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\$We can determine that:

5>4>3>1 Therefore, the correct answer is option A

Let's simplify the expression, first we'll reduce the fraction under the square root:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}=$

We'll use two exponent laws:

A. Definition of root as a power:

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

B. The power law for powers applied to terms in parentheses:

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

Let's return to the expression we received, first we'll use the law mentioned in A and convert the square root to a power:

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

We'll continue and apply the power law mentioned in B, meaning- we'll apply the power separately to the numerator and denominator, in the next step we'll remember that raising the number 1 to any power will always give the result 1, and in the fraction's denominator we'll return to the root notation, again, using the power law mentioned in A (in the opposite direction):

$\big(\frac{1}{2}\big)^{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$ Let's summarize the simplification of the given expression:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$Therefore, the correct answer is answer D.

Let's start with a reminder of the definition of a root as a power:

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

We will then use the fact that raising the number 1 to any power always yields the result 1, particularly raising it to the power of half of the square root (which we obtain by using the definition of a root as a power mentioned earlier).

In other words:

$$$\sqrt{30}\cdot\sqrt{1}= \\ \downarrow\\ \sqrt{30}\cdot\sqrt[2]{1}=\\ \sqrt{30}\cdot 1^{\frac{1}{2}}=\\ \sqrt{30} \cdot1=\\ \boxed{\sqrt{30}}$

Therefore, the correct answer is answer C.

Let's start by recalling how to define a root as a power:

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

Next, we will remember that raising 1 to any power will always yield the result 1, even the half power of the square root.

In other words:

$$$\sqrt{16}\cdot\sqrt{1}= \\ \downarrow\\ \sqrt{16}\cdot\sqrt[2]{1}=\\ \sqrt{16}\cdot 1^{\frac{1}{2}}=\\ \sqrt{16} \cdot1=\\ \sqrt{16} =\\ \boxed{4}$Therefore, the correct answer is answer D.

Let's start by recalling how to define a square root as a power:

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

Next, we remember that raising 1 to any power always gives us 1, even the half power we got from converting the square root.

In other words:

$$$\sqrt{1} \cdot \sqrt{2}= \\ \downarrow\\ \sqrt[2]{1}\cdot \sqrt{2}=\\ 1^{\frac{1}{2}} \cdot\sqrt{2} =\\ 1\cdot\sqrt{2}=\\ \boxed{\sqrt{2}}$Therefore, the correct answer is answer a.

To simplify the given expression, we will use the following three laws of exponents:

a. Definition of root as an exponent:

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

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

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

c. Law of exponents for an exponent raised to an exponent:

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

We'll start by converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{25x^4}= \\ \downarrow\\ (25x^4)^{\frac{1}{2}}=$

We'll continue, using the law of exponents mentioned in b. and apply the exponent to each factor in the parentheses:

$(25x^4)^{\frac{1}{2}}= \\ 25^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}$

We'll continue, using the law of exponents mentioned in c. and perform the exponent applied to the term with an exponent in parentheses (the second factor in the multiplication):

$25^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 25^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 25^{\frac{1}{2}}\cdot x^{2}=\\ \sqrt{25}\cdot x^2=\\ \boxed{5x^2}$

In the final steps, we first converted the power of one-half applied to the first factor in the multiplication back to the fourth root form, again, according to the definition of root as an exponent mentioned in a. (in the reverse direction) and then calculated the known fourth root of 25.

Therefore, the correct answer is answer a.

In order to solve this problem, we'll follow these steps:

• Step 1: Identify the base and exponents

• Step 2: Use the formula for multiplying powers with the same base

• Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is $(3^4) \times (3^2)$. Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
$a^m \times a^n = a^{m+n}$

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
$3^4 \times 3^2 = 3^{4+2} = 3^6$

Therefore, the simplified form of the expression is $3^6$.

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{3^5}{3^2} = 3^{5-2}$

Simplifying, we get $3^3$

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{5^6}{5^4} = 5^{6-4}$. Simplifying, we get $5^2$.

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

• Identify the given information and relevant exponent rules.

• Apply the quotient property of exponents.

• Simplify the expression.

Now, let's work through each step:
Step 1: The problem gives us the expression $\frac{6^7}{6^4}$. The base is 6, and the exponents are 7 and 4, respectively.
Step 2: According to the rule of exponents, when dividing powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ In this case, $a = 6$, $m = 7$, and $n = 4$.
Step 3: Applying this rule gives us: $\frac{6^7}{6^4} = 6^{7 - 4} = 6^3$

Therefore, the solution to the problem is $6^3$.

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

• Step 1: Identify the provided expression: $(9^2)^4$.

• Step 2: Apply the power of a power rule for exponents.

• Step 3: Simplify by multiplying the exponents.

Now, let's work through each step:

Step 1: We have the expression $(9^2)^4$.

Step 2: Using the power of a power rule ($(a^m)^n = a^{m \cdot n}$), apply it to the expression:

$(9^2)^4 = 9^{2 \times 4}$

Step 3: Simplify by calculating the product of the exponents:

$2 \times 4 = 8$

Therefore, $(9^2)^4 = 9^8$.

The correct expression corresponding to the given problem is $\boxed{9^8}$.

To solve this problem, we'll simplify the expression $2^{10} \times 3^6 \times 2^5 \times 3^2$ using the rules of exponents. Here are the steps:

• Step 1: Apply the product of powers property to the base 2 terms. The expression $2^{10} \times 2^5$ simplifies to:

  $2^{10+5} = 2^{15}$

• Step 2: Apply the product of powers property to the base 3 terms. The expression $3^6 \times 3^2$ simplifies to:

  $3^{6+2} = 3^8$

• Step 3: Combine the simplified terms to form the complete simplified expression:

  $2^{15} \times 3^8$

Therefore, the simplified form of the equation is $2^{15} \times 3^8$.

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

• Step 1: Identify and group the terms with the same base.

• Step 2: Apply the laws of exponents to simplify by adding the exponents of each base.

• Step 3: Write the simplified form.

Let's work through each step:

Step 1: We are given that $4^7 \times 5^3 \times 4^2 \times 5^4$.

Step 2: First, group the terms with the same base:

$4^7 \times 4^2$ and $5^3 \times 5^4$.

Step 3: Use the law of exponents, which states $a^m \times a^n = a^{m+n}$.

For the base 4: $4^7 \times 4^2 = 4^{7+2} = 4^9$.

For the base 5: $5^3 \times 5^4 = 5^{3+4} = 5^7$.

Therefore, the simplified form of the expression is $4^9 \times 5^7$.

To solve this problem, we'll apply the laws of exponents to simplify the expression $7^5 \times 2^3 \times 7^2 \times 2^4$.

Let's follow these steps:

• Step 1: Identify like bases.
  We have two like bases in the expression: 7 and 2.

• Step 2: Apply the product of powers rule for each base separately.
  For the base 7: $7^5 \times 7^2 = 7^{5+2} = 7^7$.
  For the base 2: $2^3 \times 2^4 = 2^{3+4} = 2^7$.

• Step 3: Combine the results.
  The expression simplifies to $7^7 \times 2^7$.

The simplified form of the original expression is therefore $7^7 \times 2^7$.

Let's simplify the expression $5^3 \times 2^4 \times 5^2 \times 2^3$ using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

• Step 1: Focus on terms with the same base.
  Combine $5^3$ and $5^2$. Since both terms have the base $5$, we apply the rule $a^m \times a^n = a^{m+n}$: $5^3 \times 5^2 = 5^{3+2} = 5^5$

• Step 2: Combine $2^4$ and $2^3$. Similarly, for the base $2$: $2^4 \times 2^3 = 2^{4+3} = 2^7$

After simplification, the expression becomes:
$5^5 \times 2^7$