Rules of Roots Combined Practice Problems & Exercises

We use the zero exponent rule.

$X^0=1$We obtain

$112^0=1$Therefore, the correct answer is option C.

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

• Step 1: Identify the given information

• Step 2: Apply the appropriate exponent rule

• Step 3: Perform the calculation

Now, let's work through each step:
Step 1: The problem gives us the expression $1^3$. This means we have a base of 1 and an exponent of 3.
Step 2: We'll use the exponentiation rule, which states that $a^n = a \times a \times \ldots \times a$ (n times).
Step 3: Since our base is 1, raising 1 to any power will still result in 1. Therefore, we can express this as $1 \times 1 \times 1 = 1$.

Therefore, the solution to $1^3$ is $1$.

To solve the problem of finding $7^0$, we will follow these steps:

• Step 1: Identify the general rule for exponents with zero.

• Step 2: Apply the rule to the given problem.

• Step 3: Consider the provided answer choices and select the correct one.

Now, let's work through each step:

Step 1: A fundamental rule in exponents is that any non-zero number raised to the power of zero is equal to one. This can be expressed as: $a^0 = 1$ where $a$ is not zero.

Step 2: Apply this rule to the problem: Since we have $7^0$, and $7$ is certainly a non-zero number, the expression evaluates to 1. Therefore, $7^0 = 1$.

Therefore, the solution to the problem is $7^0 = 1$, which corresponds to choice 2.

Simplify the following expression:

Begin by reducing the fraction under the square root:

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

Apply 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 that we obtained. Apply the law mentioned in A and convert the square root to a power:

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

Next use the power law mentioned in B, apply the power separately to the numerator and denominator.

In the next step remember that raising the number 1 to any power will always result in 1.

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.