Multiplication of Powers: Calculating powers with negative exponents

Note that we need to calculate multiplication between terms with identical bases, so we'll use the appropriate exponent law:

$b^m\cdot b^n=b^{m+n}$Note that we can only use this law to calculate multiplication performed between terms with identical bases,

Here in the problem there is also a term with a negative exponent, but this does not pose an issue regarding the use of the aforementioned exponent law. In fact, this exponent law is valid in all cases for numerical terms with different exponents, including negative exponents, rational number exponents, and even irrational number exponents, etc.,

Let's apply it to the problem:

$y^{-2}\cdot y^7=y^{-2+7}=y^5$Therefore the correct answer is A.

We begin by using the rule for multiplying exponents. (the multiplication between terms with identical bases):

$a^m\cdot a^n=a^{m+n}$We then apply it to the problem:

$7^5\cdot7^{-6}=7^{5+(-6)}=7^{5-6}=7^{-1}$When in a first stage we begin by applying the aforementioned rule and then continue on to simplify the expression in the exponent,

Next, we use the negative exponent rule:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression obtained in the previous step:

$7^{-1}=\frac{1}{7^1}=\frac{1}{7}$We then summarise the solution to the problem: $7^5\cdot7^{-6}=7^{-1}=\frac{1}{7}$Therefore, the correct answer is option B.

To solve the expression $6^3 \times 6^{-4} \times 6^7$, we need to apply the rules of exponents, specifically the multiplication of powers. When we multiply powers with the same base, we add their exponents.

First, let's identify the base and the exponents in the expression:

• The base is 6.

• The exponents are 3, -4, and 7.

Using the exponent multiplication rule, we sum the exponents:

$6^3 \times 6^{-4} \times 6^7= 6^{3 + (-4) + 7}$

So, the solution is:

$6^{3-4+7}$

To solve this problem, we'll apply the rule for multiplying powers with the same base:

• Step 1: Recognize that all terms share the base 2.

• Step 2: Apply the multiplication rule for exponents: $2^m \times 2^n = 2^{m+n}$.

• Step 3: Combine the exponents: $2^{4} \times 2^{-2} \times 2^{3}$ becomes $2^{4 + (-2) + 3}$.

According to the provided choices, the reduced expression using the property is $2^{4-2+3}$, which aligns with choice 1.

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

• Step 1: Identify that both terms have the same base, which is 4.

• Step 2: Use the exponent rule for multiplication of powers with the same base: $a^m \times a^n = a^{m+n}$.

• Step 3: Add the exponents $-2$ and $-4$.

Now, let's work through these steps:

Step 1: We have the expression $4^{-2} \times 4^{-4}$.

Step 2: Applying the exponent rule, we combine the exponents:

$4^{-2} \times 4^{-4} = 4^{-2 + (-4)}$

Therefore, our answer is $4^{-2-4}$, which matches choice 4.

To solve the problem of simplifying $2^6 \times 2^{-3}$, we follow these steps:

• Identify the problem involves multiplying powers with the same base, $2$.

• Use the formula $a^m \times a^n = a^{m+n}$ to combine the exponents.

• Add the exponents: $6 + (-3)$.

Applying the exponent rule, we calculate:

Step 1: Given expression is $2^6 \times 2^{-3}$.

Step 2: According to the property of exponents, add the exponents: $6 + (-3)$.

Step 3: Simplify the exponent: $6 - 3 = 3$.

Thus, $2^{6-3}$.

To solve this problem, we will apply the multiplication of powers rule which states that when multiplying powers with the same base, we add their exponents.

Let's begin by analyzing the given expression: $8^4 \times 8 \times 8^{-1}$.

Each term has the base 8, allowing us to use the exponent rule directly:

• First, recognize the exponents for each term: the first term $8^4$ has an exponent of 4, the second term $8$ is equivalent to $8^1$, and the third term $8^{-1}$ has an exponent of -1.
• Then, apply the formula by adding the exponents: $4 + 1 - 1$.

The resulting expression for the exponent is $8^{4+1-1}$.

Therefore, the corresponding expression to the original product is $8^{4+1-1}$.

To solve for the expression $7^{-2}\times7^{-3}\times7^5$, we will apply the exponent rule where we add the exponents when multiplying powers with the same base.

Step 1: Identify the exponents in the expression:

• First term has exponent $-2$
• Second term has exponent $-3$
• Third term has exponent $5$

Step 2: Use the exponent rule $a^m \times a^n = a^{m+n}$

We add the exponents: $-2 + (-3) + 5$.

Step 3: Calculate the sum of the exponents:

$-2 + (-3) + 5 = -2 - 3 + 5 = -5 + 5 = 0$

Therefore, the simplified expression is $7^0$.

However, the task specifically asks us to represent the step incorporating the exponent change. In this step, it should reflect as:

$7^{-2-3+5}$, indicating the addition process before simplification to 0. Let's consider the provided choices:

The correct choice from the list provided that matches our transformation is:

• Choice 4: $7^{-2-3+5}$

Hence, the expression $7^{-2}\times7^{-3}\times7^5$ can be represented by the expression $7^{-2-3+5}$.

Therefore, the correct representation is $7^{-2-3+5}$.

To solve this problem, we'll apply the rule for multiplying powers with the same base:

• Step 1: Identify the expression given, $5^{-2}\times5^{-1}\times5$.
• Step 2: Notice that all terms are powers of 5. Therefore, we can add their exponents.
• Step 3: The exponents are -2 for $5^{-2}$, -1 for $5^{-1}$, and 1 for $5$.

Let's perform the required calculations:

$-2 + (-1) + 1 = -2$

Using the power rule, the expression simplifies to:

$5^{-2} = 5^{-2}$

Therefore, the reduced form of the equation $5^{-2}\times5^{-1}\times5$ is $5^{-2}$.

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

• Step 1: Identify the common base.
• Step 2: Apply the multiplication of powers rule.
• Step 3: Simplify by adding the exponents.

Now, let's work through each step:
Step 1: The base in every term of the expression $9^{-3} \times 9^{-5} \times 9^{-2}$ is 9.
Step 2: Use the exponent multiplication rule $a^m \times a^n = a^{m+n}$.
Step 3: Add the exponents: $-3 + (-5) + (-2) = -10$.
The expression simplifies to $9^{-10}$.

Therefore, the solution to the problem is $9^{-10}$.

To simplify the equation $10 \times 10^{-3} \times 10^5$, we will apply the exponent multiplication rule which states that when multiplying like bases, we add the exponents.

• Step 1: Identify exponents on each term - The base for all terms is $10$.
• Step 2: The expression can be rewritten using implied exponents:
  $10^1 \times 10^{-3} \times 10^5$
• Step 3: Apply the rule of exponents. When multiplying terms with the same base, add the exponents:
  $10^{1 + (-3) + 5}$
• Step 4: Calculate the sum of the exponents:
  $1 + (-3) + 5 = 3$
• Step 5: Rewrite the expression with the summed exponent:
  $10^3$

Therefore, the simplified expression is $10^3$, which is choice 4.

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

• Step 1: Identify the components of the expression.
• Step 2: Apply the laws of exponents.
• Step 3: Simplify the expression by combining the powers.

Now, let's work through each step:
Step 1: The given expression is $8^{-10} \times 8^5 \times 8^4$. All the bases are the same (8).
Step 2: Using the formula $a^m \times a^n = a^{m+n}$, combine the exponents:
- The combined exponent from $-10, 5,$ and $4$ is calculated as $(-10) + 5 + 4$.
Step 3: Simplify the expression:
- Calculate the sum of the exponents: $-10 + 5 + 4 = -1$.
- This simplifies to $8^{-1}$.

Therefore, the solution to the problem is $8^{-1}$.

To solve this problem, we will simplify the expression $3^{-2} \times 3^4$ using the rules of exponents:

Step 1: Recognize that both numbers have the same base, 3. Therefore, we can apply the rule for multiplying powers of the same base, which is to add the exponents: $3^a \times 3^b = 3^{a+b}$.

Step 2: Add the exponents:

$(-2) + 4 = 2$

Step 3: Write the expression with the new exponent:

$3^2$

Thus, the simplified form of $3^{-2} \times 3^4$ is $3^2$.

The correct answer is $3^2$.

To solve the expression $4^3 \times 4^{-5}$, we need to apply the multiplication of powers rule. This rule states that when you multiply two powers with the same base, you can add their exponents. Mathematically, this is expressed as:

• $a^m \times a^n = a^{m+n}$

In our case, the base $a$ is 4, and the exponents $m$ and $n$ are 3 and -5, respectively.

Applying the rule:

$4^3 \times 4^{-5} = 4^{3 + (-5)}$

Simplifying the exponent:

$3 + (-5) = -2$

So, the expression simplifies to:

$4^{-2}$

This is the reduced form of the given equation.

To solve the problem of simplifying the expression $5^{-3} \times 5^{-4}$, we will apply the exponent multiplication rule, which states that when multiplying powers with the same base, we simply add their exponents.

• Step 1: Identify the expression and confirm both terms share the base of 5. They are $5^{-3}$ and $5^{-4}$.
• Step 2: Apply the exponent multiplication rule: $5^{-3} \times 5^{-4} = 5^{(-3) + (-4)}$.
• Step 3: Simplify the expression by adding the exponents: $-3 + (-4) = -7$.

Therefore, the expression simplifies to $5^{-7}$.

Given the possible choices, the correct answer is $5^{-7}$, which corresponds to choice (1).

Thus, the solution to the problem is $5^{-7}$.

To simplify the expression $6^{-7} \times 6^3$, we use the rule for multiplying powers with the same base, which states that we add the exponents together.

Given the expression:

$6^{-7} \times 6^3$

Step 1: Identify the base and the exponents involved. The base here is 6, with exponents $-7$ and $3$.

Step 2: Apply the multiplication of powers rule:

$a^m \times a^n = a^{m+n}$

For our problem, $a = 6$, $m = -7$, and $n = 3$. Therefore:

$6^{-7} \times 6^3 = 6^{-7 + 3}$

Step 3: Calculate the exponent:

$-7 + 3 = -4$

Therefore, the expression simplifies to:

$6^{-4}$

The solution to the equation is $6^{-4}$.

We begin by using the power rule of exponents; for the multiplication of terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply it to the given problem:

$12^4\cdot12^{-6}=12^{4+(-6)}=12^{4-6}=12^{-2}$When in a first stage we apply the aforementioned rule and then simplify the subsequent expression in the exponent,

Next, we use the negative exponent rule:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression that we obtained in the previous step:

$12^{-2}=\frac{1}{12^2}=\frac{1}{144}$Lastly we summarise the solution to the problem: $12^4\cdot12^{-6}=12^{-2} =\frac{1}{144}$Therefore, the correct answer is option A.

Let's solve the problem step-by-step:

• Step 1: Identify the base and exponents involved.
• Step 2: Apply the exponent multiplication rule to simplify.
• Step 3: Further simplify the expression if necessary.

Now, we will work through each step:

Step 1: The problem gives us the expression $6^{-5} \times 6^2$. We have a common base, which is 6.

Step 2: Using the rule for multiplying exponents with the same base, we add the exponents. Thus, $6^{-5} \times 6^2 = 6^{-5+2} = 6^{-3}$.

Step 3: Simplifying further, since a negative exponent means the reciprocal, we have:
$6^{-3} = \frac{1}{6^3}$.

Therefore, the solution to the problem is: $\frac{1}{6^3}$.

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.