Powers Special Cases Practice - Negative Exponents & Zero Power

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.

We use the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will write the fraction in parentheses as a negative power with the help of the previously mentioned power:

$\frac{1}{4}=\frac{1}{4^1}=4^{-1}$We return to the problem, where we obtained:

$\big(\frac{1}{4}\big)^{-1}=(4^{-1})^{-1}$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$(4^{-1})^{-1}=4^{-1\cdot-1}=4^1=4$Therefore, the correct answer is option d.

We use the property of powers of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the problem:

$5^{-2}=\frac{1}{5^2}=\frac{1}{25}$

Therefore, the correct answer is option d.

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 $1$.