Powers - special cases - 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.

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 $1$.

We begin by using the power rule of negative exponents.

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

$$$4^{-1}=\frac{1}{4^1}=\frac{1}{4}$We can therefore deduce that the correct answer is option B.

Using the rules of negative exponents: how to raise a number to a negative exponent:

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

$$$$$7^{-24}=\frac{1}{7^{24}}$Therefore, the correct answer is option D.

In order to solve the exercise, we use the negative exponent rule.

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

We apply the rule to the given exercise:

$19^{-2}=\frac{1}{19^2}$

We can then continue and calculate the exponent.

$\frac{1}{19^2}=\frac{1}{361}$

We use the negative exponent rule.

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

We apply it to the problem in the opposite sense.:

$$$$$\frac{1}{8^3}=8^{-3}$

Therefore, the correct answer is option A.

We use the power property for a negative exponent:

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

$\frac{1}{2^9}=2^{-9}$

Therefore, the correct answer is option A.

To begin with, we must remind ourselves of the Negative Exponent rule:

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

$\frac{1}{12^3}=12^{-3}$Therefore, the correct answer is option A.

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, we need to evaluate expressions by applying the rules of exponents and the effects of parentheses on negative numbers:

• $(-3)^2$: When a negative number is squared, the result is positive. So, $(-3)^2$ means $-3 \times -3 = 9$.
• $-(-3)^2$: This means $-1 \times (-3 \times -3)$ because squaring a number negates the negative sign inside parentheses, resulting in $-9$.
• $-(3)^2$: This equals $-1 \times (3 \times 3) = -9$, as the negative sign is outside the squared value.
• $-3$: This is simply $-3$.

Only $(-3)^2$ equals 9, confirming it as the correct expression required by the problem.

Therefore, the solution to the problem is $(-3)^2$.