Powers Special Cases Practice - Negative Exponents & Zero Power

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 $(-2)^7$, follow these steps:

• Step 1: Identify the base and the exponent given in the expression, which are $-2$ and $7$, respectively.
• Step 2: Recognize that since the exponent is $7$, which is an odd number, the result of the power will remain negative: $(-2)^7$ will be $- (2^7)$.
• Step 3: Compute $2^7$. This involves multiplying $2$ by itself $7$ times:
  $2 \times 2 = 4$
  $4 \times 2 = 8$
  $8 \times 2 = 16$
  $16 \times 2 = 32$
  $32 \times 2 = 64$
  $64 \times 2 = 128$
  Thus, $2^7 = 128$.
• Step 4: Apply the negative sign to the result of $2^7$, resulting in $-128$.

Therefore, the value of $(-2)^7$ is $-128$.

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 $4^{-2}$.