Multiplication of Powers: Presenting powers with negative exponents as fractions

To solve the expression $11^{-2} \times 11^{-5} \times 11^{-4}$, we apply the rules for multiplying numbers with the same base:

• Step 1: Use the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$.

• Step 2: Add the exponents: $-2 + -5 + -4$.

• Step 3: Perform the calculation: $-2 - 5 - 4 = -11$.

• Step 4: Write the expression with the combined exponent: $11^{-11}$.

• Step 5: Express $11^{-11}$ as a positive power using the property of negative exponents: $a^{-n} = \frac{1}{a^n}$.

Therefore, $11^{-11} = \frac{1}{11^{11}}$.

The final answer is $\frac{1}{11^{11}}$.

To solve the problem $9^{-1} \times 9^{-2} \times 9^{-3}$, we follow these steps:

• Step 1: Use the rule for multiplying exponential terms with the same base. The formula is $a^m \times a^n = a^{m+n}$.
• Step 2: Apply the formula to the given expression: $9^{-1} \times 9^{-2} \times 9^{-3} = 9^{-1-2-3}$.
• Step 3: Simplify the exponent: $-1 - 2 - 3 = -6$. Therefore, $9^{-1} \times 9^{-2} \times 9^{-3} = 9^{-6}$.

We can express $9^{-6}$ as a positive power by recalling that negative exponents indicate reciprocals:

$9^{-6} = \frac{1}{9^6}$.

Thus, both $9^{-6}$ and $\frac{1}{9^6}$ are valid expressions for the simplified form. Additionally, the expression $9^{-1-2-3}$ highlights the step where we combined the exponents, and it is equivalent to the final result. Therefore, all given answers correctly represent the simplified expression.

Therefore, the solution to the problem is All answers are correct.

To solve this problem, we need to simplify the expression $8^{-10} \times 8^{-5} \times 8^9$ using exponent rules.

• Step 1: Apply the multiplication rule for exponents. This rule states that when multiplying expressions with the same base, you add their exponents. Thus, we calculate:
  $8^{-10} \times 8^{-5} \times 8^9 = 8^{-10 + (-5) + 9}$.
• Step 2: Simplify the exponents:
  $-10 + (-5) + 9 = -10 - 5 + 9 = -6$.
• Step 3: The expression simplifies to:
  $8^{-6}$.
• Step 4: Convert the negative exponent into a positive one by using the rule for negative exponents, where $a^{-n} = \frac{1}{a^n}$:
  $8^{-6} = \frac{1}{8^6}$.

Therefore, the simplified expression is $\frac{1}{8^6}$.

The corresponding expression is:

$\frac{1}{8^6}$

To simplify the expression $4^{-6} \times 4$, follow these steps:

• Step 1: Apply the rule for multiplying powers with the same base, which is $a^m \times a^n = a^{m+n}$.

• Step 2: Identify the exponents for the terms. Here, we have $4^{-6}$ and $4^1$, implying $m = -6$ and $n = 1$.

• Step 3: Add the exponents: $(-6) + 1 = -5$. Thus, we have $4^{-6} \times 4^1 = 4^{-5}$.

• Step 4: Recognize that a negative exponent indicates a reciprocal. Therefore, $4^{-5} = \frac{1}{4^5}$.

Therefore, the solution to the expression $4^{-6} \times 4$ is $\frac{1}{4^5}$.

Let's simplify the expression $5^{-8} \times 5^6$ using the rules of exponents.

• Step 1: Apply the rule for multiplying powers with the same base. According to this rule, when multiplying like bases, we add the exponents: $5^{-8} \times 5^6 = 5^{-8 + 6}$
• Step 2: Calculate the sum of the exponents: $-8 + 6 = -2$.
• Step 3: Write the simplified expression: $5^{-2}$.
• Step 4: Relate the expression to the given choices:

The simplified expression $5^{-2}$ corresponds to choice 1. Additionally, rewriting a negative exponent using the fraction format gives: $5^{-2} = \frac{1}{5^2}$, which matches choice 2.

Thus, both choices 'a: $5^{-2}$' and 'b: $\frac{1}{5^2}$' are correct.

Therefore, according to the given answer choice, a'+b' are correct.