Multiplication of Powers: Operations in stages

To solve this problem, let’s follow the outlined steps:

• Step 1: Calculate $10 \times 10$.
• Step 2: Express the calculation using exponent rules.
• Step 3: Verify the possible expressions match the given choices.

Now, let's work through each step:
Step 1: The direct multiplication of 10 by 10 yields $100$ because $10 \times 10 = 100$.
Step 2: We can express this calculation using the rules of exponents. Since both numbers are 10 and multiplied together: $10^1 \times 10^1 = 10^{1+1} = 10^2$.
Step 3: We consider the following expressions given in the multiple-choice answers:
- Choice 1: $10^{1+1}$ equals $10^2$.
- Choice 2: $10^2$ equals 100.
- Choice 3: 100 is the result of the direct multiplication.

Each choice is consistent with the others through these steps. Thus, all the provided expressions—$10^{1+1}$, $10^2$, and 100—accurately represent the resolved equation $10 \times 10$.

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

To solve the problem of simplifying the equation $11^{10} \times 11^{11}$, follow these steps:

• Step 1: Identify that the bases are the same (11).

• Step 2: Apply the multiplication of powers rule, which states that when multiplying like bases, you add the exponents.

• Step 3: Add the exponents: $10 + 11$.

• Step 4: Perform the addition: $10 + 11 = 21$.

• Step 5: Write the expression with the new exponent: $11^{10+11}= 11^{21}$.

Therefore, the simplified expression is $11^{21}$. This corresponds to options 1 and 2 being correct as they represent the same expression when evaluating the sum, which is also represented by choice 4 as "a'+b' are correct".

To simplify the equation $12 \times 12^2$, follow these steps:

• Step 1: Recognize that 12 can be expressed as a power. Since $12 = 12^1$, rewrite the equation as $12^1 \times 12^2$.
• Step 2: Apply the rule for multiplying powers with the same base, which states that $a^m \times a^n = a^{m+n}$. In this case, this becomes $12^{1+2}$.
• Step 3: Simplify the expression by adding the exponents: $1 + 2 = 3$.

Thus, the simplified form of the expression is $12^3$.

Therefore, the correct answer choice is $12^{1+2}$, which corresponds to choice 2.

To simplify the expression $2^2 \times 2^3$, we apply the rule for multiplying powers with the same base. According to this rule, when multiplying two exponential expressions that have the same base, we keep the base and add the exponents.

• Step 1: Identify the base: In this problem, the base for both terms is 2.
• Step 2: Apply the exponent multiplication rule: $2^2 \times 2^3 = 2^{2+3}$.
• Step 3: Simplify by adding the exponents: $2^{2+3} = 2^5$.

Thus, the simplified form of the expression $2^2 \times 2^3$ is $2^{5}$.

The correct choice from the provided options is: $2^{2+3}$.

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

• Step 1: Identify the given expression and its components.

• Step 2: Apply the exponent multiplication formula.

• Step 3: Simplify the result.

Now, let's work through each step:
Step 1: The given expression is $3^4 \times 3^5$. We recognize that the base is 3 and the exponents are 4 and 5.
Step 2: Apply the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$. Using this formula, we add the exponents: $4 + 5$.
Step 3: Simplify the expression: $3^{4+5} = 3^9$.

Therefore, the simplified form of the expression is $3^{4+5}$.

To simplify the expression $5^8 \times 5^3$, we will use the exponent rule which states that when multiplying powers with the same base, we add the exponents:

Step-by-step:

• Identify the base and exponents: Here, the base is $5$, and the exponents are $8$ and $3$.

• Apply the multiplication of exponents rule: $5^8 \times 5^3 = 5^{8+3}$.

Therefore, the correct answer is $5^{8+3}$.

To simplify the expression given, $6^2 \times 6^8$, we will use the property of exponents which states that the product of two powers with the same base is the base raised to the sum of the exponents.

Let's apply the rule:

• The base in both powers is $6$.

• The exponents are $2$ and $8$.

• According to the rule $a^m \times a^n = a^{m+n}$, we add the exponents; therefore, $6^2 \times 6^8 = 6^{2+8}$.

• Simplifying further, this becomes $6^{10}$.

Therefore, the simplified expression is $6^{10}$.

The solution to the given problem is $6^{2+8}$.

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

• Step 1: Identify the given expression.

• Step 2: Recognize and apply the exponent multiplication rule.

• Step 3: Simplify the expression by adding the exponents.

Now, let's work through each step:
Step 1: The expression given is $7^6 \times 7^6$.
Step 2: Since the bases are the same, apply the exponent rule: $a^m \times a^n = a^{m+n}$.
Step 3: By adding the exponents, we have $6 + 6 = 12$.

Therefore, the simplified expression is $7^{6+6}$ or $7^{12}$.

This corresponds to choice 2.

Thus, the solution to the problem is $7^{6+6}$.

To simplify the expression $8^3 \times 8$, we begin by identifying the implicit exponent for the standalone 8. Since there is no written exponent next to the second 8, we can assume it has an exponent of 1.

Thus, the expression can be written as:\br $8^3 \times 8^1$.

Using the rule for multiplying powers with the same base, $a^m \times a^n = a^{m+n}$, we add the exponents:

• Here, the base $a$ is 8.
• The exponents are 3 and 1.

Therefore, $8^3 \times 8^1 = 8^{3+1} = 8^4$.

Thus, the simplified expression is $\mathbf{8^4}$.

Consequently, the correct choice is $8^{3+1}$.

To solve this problem, let's apply the multiplication of powers rule:

• Step 1: Identify expression as $9 \times 9^9$.
• Step 2: Note that $9$ can be expressed as $9^1$.
• Step 3: Apply the exponent rule: $a^m \times a^n = a^{m+n}$.

Now, we'll work through the calculation step-by-step:

Step 1: Rewrite $9$ as $9^1$. Thus, our expression becomes $9^1 \times 9^9$.

Step 2: Use the exponent rule to combine: $9^1 \times 9^9 = 9^{1+9}$.

Step 3: Simplify the exponent by adding: $9^{1+9} = 9^{10}$.

Therefore, the simplified form of the expression is $9^{10}$.

In terms of the answer choices, the correct answer is

$9^{1+9}$

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

Let's go through each step:

• Identify the expression: $10^5 \times 10^7 \times 10^2$.

• Notice that the base for all terms is 10, so we apply the product of powers rule: $a^m \times a^n = a^{m+n}$.

• Add the exponents: $5 + 7 + 2$.

Now, calculate the sum of the exponents:

$5 + 7 + 2 = 14$.

Therefore, according to the rule, the expression simplifies to:

$10^{5+7+2}=10^{14}$.

To solve this problem, we will simplify the expression $11^2 \times 11^3 \times 11^4$ by using the multiplication rule of exponents.

• Step 1: Identify that all the bases are the same, which is 11.

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

Now, apply this rule:

$11^2 \times 11^3 \times 11^4 = 11^{2+3+4}$

Calculate the sum of the exponents:

$2 + 3 + 4 = 9$

Thus, the expression simplifies to:

$11^9$

Therefore, the simplified version of the expression is:

$11^{2+3+4} = 11^9$

Upon reviewing the choices provided, the correct choice for the simplified expression is choice 3: $11^{2+3+4}$.

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

• Step 1: Identify the given expression $20^6 \times 20^2 \times 20^4$.
• Step 2: Apply the rule of exponents for multiplication of like bases, which is $a^m \times a^n = a^{m+n}$.
• Step 3: Add the exponents together to simplify the power expression.

Now, let's work through each step:
Step 1: We have $20^6 \times 20^2 \times 20^4$.
Step 2: Apply the property of exponents: $20^6 \times 20^2 \times 20^4 = 20^{6+2+4}$.
Step 3: Add the exponents: $6 + 2 + 4 = 12$, so the expression simplifies to $20^{12}$.

By checking the given choices, the correct one is:

Choice 4: A'+C' are correct

To simplify the expression $2^1 \times 2^2 \times 2^3$, we'll apply the rule for multiplying powers with the same base:

• When multiplying powers with the same base, you add the exponents.

Let's apply this to our expression:

$2^1 \times 2^2 \times 2^3 = 2^{1+2+3}$

Now, calculate the sum of the exponents: $1 + 2 + 3 = 6$.

Thus, the expression simplifies to

$2^6$.

By comparing it with the given choices, the correct simplified form, $2^{1+2+3}$, corresponds to choice 2:
$2^{1+2+3}$.

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

• Step 1: Identify the exponents in the expression.
• Step 2: Use the rule for multiplying powers with the same base.
• Step 3: Simplify the expression with the combined exponent.

Now, let's work through each step:

Step 1: From the expression $-4^3 \times -4^4 \times -4^2$, the exponents of $-4$ are 3, 4, and 2.

Step 2: Using the formula for multiplying powers with the same base, which is $a^m \times a^n = a^{m+n}$, add the exponents: $3 + 4 + 2 = 9$.

Step 3: Rewrite the expression using the combined exponent: $-4^{3 + 4 + 2} = -4^9$.

Therefore, the simplified form of the given expression is $-4^9$.

The correct answer to the problem is indeed $-4^9$, which matches choice (3) in the provided options.

To solve this simplification problem, we will apply the rules of exponents. Our steps are as follows:

• Step 1: Identify the exponents of the base. We have $4^5$, $4^1$ (since $4$ is equivalent to $4^1$), and $4^2$.
• Step 2: Use the property of exponents: $a^m \times a^n = a^{m+n}$ to combine powers of the same base.
• Step 3: Calculate the sum of the exponents: $5 + 1 + 2 = 8$.
• Step 4: Express the simplified result in the form of a single power of 4: $4^{8}$.

Therefore, the expression $4^5 \times 4 \times 4^2$ simplifies to $4^{5+1+2}$, which further simplifies to $4^8$.

Checking the multiple-choice options, the correct choice is: $4^{5+1+2}$, aligning with our solution.

To solve the problem of simplifying the expression $5^3 \times 5^6 \times 5^2$, follow these steps:

• Step 1: Understand that the expression involves multiplying powers with the same base.

• Step 2: Apply the formula for multiplying powers: $a^m \times a^n = a^{m+n}$.

• Step 3: Combine the exponents by adding them together.

Now, let's work through these steps in detail:
Step 1: Recognize the base is 5, with exponents 3, 6, and 2.
Step 2: Since all terms have the base 5, use the formula for multiplying powers, resulting in a single term where the exponents are added: $5^{3+6+2}$.
Step 3: Calculate the sum of the exponents: $3 + 6 + 2 = 11$.

Hence, the correct answer is $5^{3+6+2}$ which simplifies to $5^{11}$.

To simplify the expression $9^7 \times 9^3 \times 9^5$, we will use the multiplication rule for exponents which applies to powers with the same base.

• Step 1: Identify that all terms have the same base of 9.
• Step 2: Add the exponents together since the bases are the same: $7 + 3 + 5$.
• Step 3: Perform the addition: $7 + 3 + 5 = 15$.

This results in the expression simplifying to $9^{7+3+5} = 9^{15}$.

Therefore, the expression $9^7 \times 9^3 \times 9^5$ simplifies to $9^{15}$.

The correct answer is choice (1): $9^{7+3+5}$.

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 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, $2^{6-3}$.