Negative Exponents: Presenting powers with negative exponents as fractions

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 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 $3^4$.