Product Property of Square Roots - Examples, Exercises and Solutions

To solve this problem, we can use the product property of square roots.

• Step 1: Recognize the expression $\sqrt{a} \cdot \sqrt{b}$.
• Step 2: Apply the product property: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.

This tells us that the original expression, $\sqrt{a} \cdot \sqrt{b}$, simplifies to $\sqrt{a \cdot b}$.

Thus, the equivalent expression is $\sqrt{a \cdot b}$.

Among the given choices, choice 2 $\sqrt{a\cdot b}$ is the correct one.

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.

To solve the expression $\sqrt{1} \cdot \sqrt{25}$, we will use the Product Property of Square Roots.

According to the property, we have:

$\sqrt{1} \cdot \sqrt{25} = \sqrt{1 \cdot 25}$

First, calculate the product inside the square root:

$1 \cdot 25 = 25$

Now the expression simplifies to:

$\sqrt{25}$

Finding the square root of 25 gives us:

$5$

Thus, the value of $\sqrt{1} \cdot \sqrt{25}$ is $\boxed{5}$.

After comparing this solution with the provided choices, we see that the correct answer is choice 3.

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.

In order to simplify the given expression, apply 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:

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

Begin 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.

To simplify the given expression, we use two laws of exponents:

A. Defining the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$B. The law of exponents for dividing powers with the same base (in the opposite direction):

$x^n\cdot y^n =(x\cdot y)^n$

Let's start by using the law of exponents shown in A:

$\sqrt{10}\cdot\sqrt{3}= \\ \downarrow\\ 10^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=$We continue, since we have a multiplication between two terms with equal exponents, we can use the law of exponents shown in B and combine them under the same base which is raised to the same exponent:

$10^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (10\cdot3)^{\frac{1}{2}}=\\ 30^{\frac{1}{2}}=\\ \boxed{\sqrt{30}}$In the last steps, we performed the multiplication of the bases and used the definition of the root as an exponent shown earlier in A (in the opposite direction) to return to the root notation.

Therefore, the correct answer is B.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start by converting the square roots to exponents using the law mentioned in a:

$\sqrt{7}\cdot\sqrt{7}= \\ \downarrow\\ 7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}=$We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b:

$7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}= \\ 7^{\frac{1}{2}+\frac{1}{2}}=\\ 7^1=\\ \boxed{7}$Therefore, the correct answer is answer a.

We can simplify the expression without using the laws of exponents, because the expression has known square roots, so let's simplify the expression and then perform the multiplication:

$\sqrt{100}\cdot\sqrt{25}=\\ 10\cdot5=\\ \boxed{50}$Therefore, the correct answer is answer D.

We can simplify the expression directly without using the laws of exponents, since the expression has known square roots, so let's simplify the expression and then perform the multiplication:

$\sqrt{25}\cdot\sqrt{4}=\\ 5\cdot2=\\ \boxed{10}$Therefore, the correct answer is answer C.

We can simplify the expression without using the laws of exponents, since the expression has known square roots, so let's simplify the expression and then perform the multiplication:

$\sqrt{9}\cdot\sqrt{4}=\\ 3\cdot2=\\ \boxed{6}$Therefore, the correct answer is answer B.

In order to simplify the given expression, apply 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:

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

Begin by converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{100x^2}= \\ \downarrow\\ (100x^2)^{\frac{1}{2}}=$

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

$(100x^2)^{\frac{1}{2}}= \\ 100^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

We'll continue, once again 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):

$100^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 100^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 100^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{100}\cdot x=\\ \boxed{10x}$

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 reverse) and then calculated the known fourth root of 100.

Therefore, the correct answer is answer d.

In order to simplify the given expression, apply 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 with converting the fourth root to an exponent using the law of exponents mentioned in a.:

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

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

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

We'll once again 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):

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

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 reverse) and then calculated the known fourth root of 49.

Therefore, the correct answer is answer c.

In order to simplify the given expression, apply 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.:

$\sqrt{16x^2}= \\ \downarrow\\ (16x^2)^{\frac{1}{2}}=$

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

$(16x^2)^{\frac{1}{2}}= \\ 16^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

We'll once again 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):

$16^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 16^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 16^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{16}\cdot x=\\ \boxed{4x}$

In the final steps, first we 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 opposite direction) and then we calculated the known fourth root of 16.

Therefore, the correct answer is answer d.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start by converting the square roots to exponents using the law mentioned in a:

$\sqrt{5}\cdot\sqrt{5}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot5^{\frac{1}{2}}=$We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b:

$5^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= \\ 5^{\frac{1}{2}+\frac{1}{2}}=\\ 5^1=\\ \boxed{5}$Therefore, the correct answer is answer a.