Powers of a Fraction: Presenting powers in the denominator as powers with negative exponents

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 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.