Powers - special cases - Examples, Exercises and Solutions

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.

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

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.

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 simplify the expression $\left(\frac{1}{20}\right)^{-7}$, we will apply the rule for negative exponents. The key idea is that a negative exponent indicates taking the reciprocal and converting the exponent to a positive:

• Start with the expression: $\left(\frac{1}{20}\right)^{-7}$.
• Apply the negative exponent rule: $\left(\frac{1}{a}\right)^{-n} = a^n$.
• For our expression: $\left(\frac{1}{20}\right)^{-7}$ becomes $20^7$.

Therefore, $\left(\frac{1}{20}\right)^{-7}$ simplifies to $20^7$.

Thus, the correct answer is $20^7$.

To solve for $\left(\frac{1}{60}\right)^{-4}$, we apply the rule for negative exponents.

Step 1: Use the negative exponent rule: For any non-zero number $a$, $a^{-n} = \frac{1}{a^n}$. Thus,

$\left(\frac{1}{60}\right)^{-4} = \left(\frac{60}{1}\right)^4$.

Step 2: Simplify $\left(\frac{60}{1}\right)^4$ by recognizing the identity $\frac{60}{1} = 60$, so it follows that:

$\left(\frac{60}{1}\right)^4 = 60^4$.

Therefore, the simplified expression is $60^4$.

The correct answer is $60^4$

To solve the given problem, we need to express $\frac{1}{5^2}$ using negative exponents. We'll apply the formula for negative exponents, which is $\frac{1}{a^n} = a^{-n}$:

• Identify the base and power in the denominator. Here, the base is $5$ and the power is $2$.
• Apply the inverse formula: $\frac{1}{5^2} = 5^{-2}$.

Thus, the equivalent expression for $\frac{1}{5^2}$ using a negative exponent is $5^{-2}$.

To solve the problem of expressing $\frac{1}{4^2}$ using powers with negative exponents:

• Identify the base in the denominator: 4 raised to the power 2.
• Apply the rule for negative exponents that states $\frac{1}{a^n} = a^{-n}$.
• Express $\frac{1}{4^2}$ as $4^{-2}$.

Thus, the expression $\frac{1}{4^2}$ can be rewritten as $4^{-2}$.

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

• Step 1: Identify the given expression with a negative exponent.
• Step 2: Apply the rule for negative exponents, which allows us to convert the expression into a positive exponent form.
• Step 3: Perform the calculation of the new expression.

Now, let's work through each step:

Step 1: The expression given is $\left(\frac{1}{3}\right)^{-4}$, which involves a negative exponent.

Step 2: According to the exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$, we can rewrite the expression with a positive exponent by inverting the fraction:

$\left(\frac{1}{3}\right)^{-4} = \left(\frac{3}{1}\right)^4 = 3^4$.

Step 3: Calculate $3^4$.

The calculation $3^4$ is as follows:

$3^4 = 3 \times 3 \times 3 \times 3 = 81$.

However, since the problem specifically asks for the corresponding expression before calculation to numerical form, the answer remains $3^4$.

Therefore, the answer to the problem, in terms of an equivalent expression, is $3^4$.

To solve this problem, we'll use the rule of negative exponents:

• Step 1: Identify that the given expression is $\frac{1}{3^2}$.
• Step 2: Recognize that $\frac{1}{3^2}$ can be rewritten using the negative exponent rule.
• Step 3: Apply the formula $\frac{1}{a^n} = a^{-n}$ to the expression $\frac{1}{3^2}$.

Now, let's work through these steps:

Step 1: We have $\frac{1}{3^2}$ where 3 is the base and 2 is the exponent.

Step 2: Using the formula, convert the denominator $3^2$ to $3^{-2}$.

Step 3: Thus, $\frac{1}{3^2} = 3^{-2}$.

Therefore, the solution to the problem is $3^{-2}$.

To solve this problem, we will rewrite the expression $\frac{1}{6^7}$ using the rules of exponents:

Step 1: Identify the given fraction.

We start with $\frac{1}{6^7}$, where the base in the denominator is 6, and the exponent is 7.

Step 2: Apply the formula for negative exponents.

The formula $a^{-n} = \frac{1}{a^n}$ allows us to rewrite a reciprocal power as a negative exponent. This means the expression $\frac{1}{6^7}$ can be rewritten as $6^{-7}$.

Step 3: Conclude with the answer.

By transforming $\frac{1}{6^7}$ to its equivalent form using negative exponents, the expression becomes $6^{-7}$.

Therefore, the correct expression is $\boxed{6^{-7}}$, which corresponds to choice 2 in the given options.

To solve this problem, we will use the properties of exponents. Specifically, we will convert the expression $\frac{1}{20^2}$ into a form that uses a negative exponent. The general relationship is that $\frac{1}{a^n} = a^{-n}$.

Applying this rule to the given expression:

• Step 1: Identify the current form, which is $\frac{1}{20^2}$.
• Step 2: Apply the negative exponent rule: $\frac{1}{20^2} = 20^{-2}$.
• Step 3: This expression, $20^{-2}$, represents $\frac{1}{20^2}$ using a negative exponent.

Therefore, the expression $\frac{1}{20^2}$ can be expressed as $20^{-2}$, which aligns with choice 1.

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.

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.

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.