Zero Exponent Rule: Applying the formula

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

• Step 1: Recognize that the base of the exponent is 1.
• Step 2: Apply the Zero Exponent Rule.
• Step 3: Verify the result is consistent with mathematical rules.

Now, let's work through each step:
Step 1: We have the expression $1^0$, where 1 is the base.
Step 2: According to the Zero Exponent Rule, any non-zero number raised to the power of zero is equal to 1. Hence, $1^0 = 1$.
Step 3: Verify: The base 1 is indeed non-zero, confirming that the zero exponent rule applies.

Therefore, the value of $1^0$ is $1$.

To solve this problem, we need to find the value of $4^0$.

• Step 1: According to the properties of exponents, for any non-zero number $a$, the zero power $a^0$ is always equal to 1.

• Step 2: Here, our base is 4, which is a non-zero number.

• Step 3: Applying the zero exponent rule, we find:

$4^0 = 1$

Thus, the answer to the question is $1$, corresponding to choice 3.

To solve the problem, $(\frac{1}{8})^0$, we utilize the Zero Exponent Rule, which states that any non-zero number raised to the power of zero equals $1$.

Here's a step-by-step explanation:

• Step 1: Identify the base and ensure it is non-zero. In this case, the base is $\frac{1}{8}$, which is indeed non-zero.
• Step 2: Apply the Zero Exponent Rule. According to this rule, $\left(\frac{1}{8}\right)^0 = 1$.
• Step 3: Conclude the result: The expression evaluates to $1$.

Therefore, the correct answer to the problem $(\frac{1}{8})^0$ is $1$.

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, let's analyze the expression $(\frac{1}{5})^0$.

• Step 1: Identify the base and exponent
  The base is $\frac{1}{5}$, and the exponent is $0$.
• Step 2: Apply the zero exponent rule
  The zero exponent rule states that any non-zero number raised to the power of zero is $1$. This rule applies universally to all real numbers except zero.
• Conclusion
  Using the rule, $(\frac{1}{5})^0 = 1$.

Therefore, the value of the expression $(\frac{1}{5})^0$ is $1$. Thus, the correct answer is choice $2$.

To solve the problem of evaluating $0^0$, we need to analyze the properties of exponents and related mathematical principles:

• Typically, for any number $b$, the expression $b^0 = 1$. However, $b^0$ assumes $b \neq 0$. When $b$ is zero, this rule conflicts with the intuitive case that would suggest $0^n = 0$ for any positive integer $n$.

• In mathematics, $0^0$ arises in contexts where it could be considered both zero and one depending on the operation taken to the limit in functions. For example, evaluating limits involving forms like $(x^x)$ as $x \to 0$ can show indeterminacy.

• Thus, $0^0$ is not defined within the normal arithmetic rules we apply to exponents because it does not yield a consistent value across mathematical contexts. Historically, it is generally considered indeterminate.

Therefore, $0^0$ is not defined.

To solve this problem, let's apply the Zero Exponent Rule:

• Step 1: Identify the base, which is $0.1$, a non-zero number.
• Step 2: Recognize that the exponent is $0$.
• Step 3: Apply the zero exponent rule, which states that any non-zero number raised to the power of zero equals $1$.

According to the Zero Exponent Rule, we have:

$(0.1)^0 = 1$.

Therefore, the value of $(0.1)^0$ is $\mathbf{1}$.

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

• Step 1: Identify the given expression which is $\frac{1}{5^0}$.
• Step 2: Recognize that $5^0$ involves a zero exponent.
• Step 3: Apply the zero exponent rule, which states that any non-zero number raised to the power of zero equals one.
• Step 4: Substitute the result back into the original expression.

Now, let's work through each step:
Step 1: The problem gives us $\frac{1}{5^0}$.
Step 2: We apply the zero exponent rule to the term $5^0$. According to the rule, $5^0 = 1$.
Step 3: Substitute this value into the expression, thus yielding $\frac{1}{1}$.
Step 4: Perform the division, which results in $1$.

Therefore, the solution to the problem is 1.

To solve the problem $\frac{2^0}{3}$, follow these steps:

• Step 1: Identify the given expression – The numerator is $2^0$.
• Step 2: Apply the Zero Exponent Rule – By the rule $a^0 = 1$ for any non-zero number $a$, we have $2^0 = 1$.
• Step 3: Simplify the expression – Replace $2^0$ with 1 to get $\frac{1}{3}$.

Therefore, the value of the expression $\frac{2^0}{3}$ is $\frac{1}{3}$.

We use the zero exponent rule.

$X^0=1$We obtain:

$\big( \frac{7}{125}\big)^0=1$Therefore, the correct answer is option B.

Due to the fact that raising any number (except zero) to the power of zero will yield the result 1:

$X^0=1$It is thus clear that:

$(\frac{7}{4})^0=1$Therefore, the correct answer is option C.

We use the law of exponents to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the law to given the problem:

$7^x\cdot7^{-x}=7^{x+(-x)}=7^{x-x}=7^0$In the first stage we apply the above power rule and in the following stages we simplify the expression obtained in the exponent,

Subsequently, we use the zero power rule:

$X^0=1$We obtain:

$7^0=1$Lastly we summarize the solution to the problem.

$7^x\cdot7^{-x}=7^{x-x}=7^0 =1$Therefore, the correct answer is option B.

First, let's note that the first term in the multiplication in the problem has an exponent of 0, and any number (different from zero) raised to the power of zero equals 1, meaning:

$X^0=1$Therefore, we get that the expression in the problem is:

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}= 1\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}=(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}$

Later, we will use the law of exponents for negative exponents:

$a^{-n}=\frac{1}{a^n}$

And before we proceed to solve the problem let's understand this law in a slightly different, indirect way:

Let's note that if we treat this law as an equation (which it indeed is in every way), and multiply both sides of the equation by the common denominator which is:

$a^n$we get:

$a^{-n}=\frac{1}{a^n}\\ \frac{a^n}{1} =\frac{1}{a^n}\hspace{8pt} \text{/}\cdot a^n\\ a^n\cdot a^{-n}=1$

Where in the first stage we remembered that any number can be represented as itself divided by 1, we applied this to the left side of the equation, then we multiplied by the common denominator and to know by how much we multiplied each numerator (after reduction with the common denominator) we addressed the question "by how much did we multiply the current denominator to get the common denominator?".

Let's look at the result we got:

$a^n\cdot a^{-n}=1$

Meaning that $a^n,\hspace{4pt}a^{-n}$are reciprocal numbers, or in other words:

$a^n$ is reciprocal to $a^{-n}$ (and vice versa),

and specifically:

$a,\hspace{4pt}a^{-1}$ are reciprocal to each other,

We can apply this understanding to the problem if we also remember that the reciprocal of a fraction is obtained by switching the numerator and denominator, meaning that the fractions:

$\frac{a}{b},\hspace{4pt}\frac{b}{a}$ are reciprocal fractions - which can be easily understood, since their multiplication will clearly give the result 1,

And if we combine this with the previous understanding, we can easily conclude that:

$\big(\frac{a}{b}\big)^{-1}=\frac{b}{a}$

In other words, raising a fraction to the power of negative one will give a result that is the reciprocal fraction, obtained by switching the numerator and denominator.

Let's return to the problem, the expression we got in the last stage is:

$(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}$

We'll use what was explained earlier and note that the fraction in parentheses in the second term of the multiplication is the reciprocal fraction to the fraction in parentheses in the first term of the multiplication, meaning that:$\frac{13}{2}= \big(\frac{2}{13} \big)^{-1}$Therefore we can simply calculate the expression we got in the last stage by converting to a common base using the above understanding:

$(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5} = (\frac{2}{13})^{-2}\cdot\big( (\frac{2}{13})^{-1}\big)^{-5}$

Where we actually just replaced the fraction in parentheses in the second term of the multiplication with its reciprocal raised to the power of negative one as mentioned earlier,

Next we'll recall the law of exponents for power of a power:

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

And we'll apply this law to the expression we got in the last stage:

$(\frac{2}{13})^{-2}\cdot\big( (\frac{2}{13})^{-1}\big)^{-5} =(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{(-1)\cdot(-5)}=(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5}$

Where in the first stage we applied the above law of exponents and then simplified the expression that resulted,

Let's summarize the solution of the problem so far, we got that:

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}= (\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5} = (\frac{2}{13})^{-2}\cdot\big( (\frac{2}{13})^{-1}\big)^{-5} =(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5}$

In the next stage we'll recall the law of exponents for multiplication of terms with identical bases:

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

We'll apply this law to the expression we got in the last stage:

$(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5} =(\frac{2}{13}\big)^{-2+5}=(\frac{2}{13}\big)^{3}$

Where in the first stage we applied the above law of exponents and then simplified the expression,

Let's summarize the solution of the problem so far, we got that:

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}= (\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5} =(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5}=(\frac{2}{13}\big)^{3}$

Therefore the correct answer is answer A.

To solve the problem $300^{-4} \cdot \left(\frac{1}{300}\right)^{-4}$, let's follow these steps:

• Step 1: Simplify the reciprocal power expression.
  The expression $\left(\frac{1}{300}\right)^{-4}$ can be simplified using the rule for reciprocals, which states $\left(\frac{1}{a}\right)^{-n} = a^n$. Thus, $\left(\frac{1}{300}\right)^{-4} = 300^4$.
• Step 2: Combine the powers.
  Now the expression becomes $300^{-4} \cdot 300^4$. Using the rule $a^m \cdot a^n = a^{m+n}$, we have $300^{-4+4} = 300^0$.
• Step 3: Calculate the final result.
  By the identity $a^0 = 1$, any non-zero number raised to the power of zero equals 1. Therefore, $300^0 = 1$.

Thus, the solution to the problem is $\boxed{1}$.

Due to the fact that raising any number (except zero) to the power of zero will give the result 1:

$X^0=1$Let's examine the expression of the problem:

$(300\cdot\frac{5}{3}\cdot\frac{2}{7})^0$The expression inside of the parentheses is clearly not 0 (it can be calculated numerically and verified)

Therefore, the result of raising to the power of zero will give the result 1, that is:

$(300\cdot\frac{5}{3}\cdot\frac{2}{7})^0 =1$Therefore, the correct answer is option A.

This problem can be solved using the Law of exponents power rules for a negative power, power over a power, as well as the power rule for the product between terms with identical bases.

However we prefer to solve it in a quicker way:

To this end, the power by power law is applied to the parentheses in which the terms are multiplied, but in the opposite direction:

$x^n\cdot y^n=(x\cdot y)^n$Since in the expression in the problem there is a multiplication between two terms with identical powers, this law can be used in its opposite sense.

$5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4$Since the multiplication in the given problem is between terms with the same power, we can apply this law in the opposite direction and write the expression as the multiplication of the bases of the terms in parentheses to which the same power is applied.

We continue and simplify the expression inside of the parentheses. We can do it quickly if inside the parentheses there is a multiplication between two opposite numbers, then their product will give the result: 1, All of the above is applied to the problem leading us to the last step:

$\big(5\cdot\frac{1}{5}\big)^4 = 1^4=1$We remember that raising the number 1 to any power will always give the result: 1, which means that:

$1^x=1$Summarizing the steps to solve the problem, we obtain the following:

$5^4\cdot(\frac{1}{5})^4=\big(5\cdot\frac{1}{5}\big)^4 =1$Therefore, the correct answer is option b.