Zero Exponent Rule - Examples, Exercises and Solutions

Let's solve the problem step by step using the Zero Exponent Rule, which states that any non-zero number raised to the power of 0 is equal to 1.

• Consider the expression: $100^0$.
• According to the Zero Exponent Rule, if we have any non-zero number, say $a$, then $a^0 = 1$.
• Here, $a = 100$ which is clearly a non-zero number, so following the rule, we find that:
• $100^0 = 1$.

Therefore, the expression $100^0$ is equivalent to 1.

We use the power property:

$X^0=1$We apply it to the problem:

$5^0=1$Therefore, the correct answer is C.

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 formula:

$a^0=1$

$\frac{9\times3}{8^0}=\frac{9\times3}{1}=9\times3$

We know that:

$9=3^2$

Therefore, we obtain:

$3^2\times3=3^2\times3^1$

We use the formula:

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

$3^2\times3^1=3^{2+1}=3^3$

When raising any number to the power of 0 it results in the value 1, mathematically:

$X^0=1$

Apply this to both the numerator and denominator of the fraction in the problem:

$\frac{4^0\cdot6^7}{36^4\cdot9^0}=\frac{1\cdot6^7}{36^4\cdot1}=\frac{6^7}{36^4}$

Note that -36 is a power of the number 6:

$36=6^2$

Apply this to the denominator to obtain expressions with identical bases in both the numerator and denominator:

$\frac{6^7}{36^4}=\frac{6^7}{(6^2)^4}$

Recall the power rule for power of a power in order to simplify the expression in the denominator:

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

Recall the power rule for division between terms with identical bases:

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

Apply these two rules to the expression that we obtained above:

$\frac{6^7}{(6^2)^4}=\frac{6^7}{6^{2\cdot4}}=\frac{6^7}{6^8}=6^{7-8}=6^{-1}$

In the first stage we applied the power of a power rule and proceeded to simplify the expression in the exponent of the denominator term. In the next stage we applied the second power rule - The division rule for terms with identical bases, and again simplified the expression in the resulting exponent.

Finally we'll use the power rule for negative exponents:

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

We'll apply it to the expression that we obtained:

$6^{-1}=\frac{1}{6}$

Let's summarize the various steps of our solution:

$\frac{4^0\cdot6^7}{36^4\cdot9^0}=\frac{1}{6}$

Therefore the correct answer is A.