Multiplication of Powers: Applying the formula

To solve the exercise we use the property of multiplication of powers with the same bases:

$a^n * a^m = a^{n+m}$

With the help of this property, we can add the exponents.

$4^2\times4^4=4^{4+2}=4^6$

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 solve this problem, we'll simplify the expression $9^2 \times 9^9$ using the multiplication of powers rule:

• Step 1: Identify the base and the exponents. The base here is 9, and the exponents are 2 and 9.
• Step 2: Apply the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$.
• Step 3: Add the exponents: $9^2 \times 9^9 = 9^{2+9} = 9^{11}$.

The expression simplifies to $9^{11}$.

Therefore, the simplified expression is $9^{11}$, which matches choice 1.

To solve this problem, we'll use the properties of exponents to simplify the expression:

• Step 1: Identify the bases and exponents in the expression $8^3 \times 8^6$.
• Step 2: Apply the product of powers property, which states $a^m \times a^n = a^{m+n}$ when the bases are the same.
• Step 3: Add the exponents: $3 + 6 = 9$.

Now, let's work through these steps:

Step 1: Both terms, $8^3$ and $8^6$, have the same base, 8.

Step 2: According to the product of powers property, we add the exponents: $8^{3+6}$.

Step 3: Simplifying the exponents gives us $8^9$.

Therefore, the simplified expression is $8^9$.

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

• Step 1: Identify the given expression
• Step 2: Apply the exponent rule for multiplication of powers with the same base
• Step 3: Simplify the expression by adding the exponents

Now, let's work through each step:
Step 1: The given expression is $6^5 \times 6^7$. Here, the base is 6, and the exponents are 5 and 7.
Step 2: We apply the exponent rule, which states that when multiplying two powers with the same base, we add the exponents. Therefore, we have:

$6^5 \times 6^7 = 6^{5+7}$

Step 3: Add the exponents: $5 + 7 = 12$. Thus, the expression simplifies to:

$6^{12}$

Therefore, the solution to the problem is $6^{12}$.

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

• Step 1: Identify the base and exponents
• Step 2: Apply the exponent multiplication rule
• Step 3: Perform the calculations

Let's work through each step:
Step 1: We have $3^2$ and $3^3$. Both have the same base, which is 3.
Step 2: According to the exponent multiplication rule $a^m \times a^n = a^{m+n}$, we add the exponents:
$2 + 3 = 5$.
Step 3: Rewrite the expression as a single power:
$3^2 \times 3^3 = 3^{2+3} = 3^5$.

Therefore, the simplified expression is $\boldsymbol{3^5}$, which corresponds to choice 2.

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 simplify the expression $7^4 \times 7$, we follow these steps:

• Step 1: Recognize that $7$ is equivalent to $7^1$. Thus, our expression becomes $7^4 \times 7^1$.
• Step 2: Apply the rule of multiplying powers with the same base, which states: $a^m \times a^n = a^{m+n}$.
• Step 3: According to the rule, add the exponents of the same base: $4 + 1$.
• Step 4: Simplify the result, yielding $7^{4+1} = 7^5$.

Thus, the simplified expression is $7^5$.

To simplify the expression $4^5 \times 4^5$, we will use the rule of exponents that states when multiplying two powers with the same base, you can add the exponents. This rule can be expressed as:

• $a^m \times a^n = a^{m+n}$

In this equation, both terms $4^5$ have the same base $4$.

According to the multiplication of powers rule:

• $4^5 \times 4^5 = 4^{5+5}$

Now, simply add the exponents:

$4^{5+5} = 4^{10}$

The simplified form of $4^5 \times 4^5$ is therefore $4^{10}$.

To solve the problem of simplifying $5 \times 5^8$, we use the rules of exponents:

• Identify that $5$ can be rewritten as $5^1$.
• Apply the multiplication of powers rule: $a^m \times a^n = a^{m+n}$.
• Add the exponents: $1 + 8 = 9$.
• Thus, $5^1 \times 5^8 = 5^{1+8} = 5^9$.

Therefore, the simplified form of the given expression is $5^9$.

Hence, the correct answer is choice : $5^9$.

To solve the problem of simplifying $15^2 \times 15^4$, we will use the rule for multiplying exponents with the same base.

According to the multiplication of powers rule: If $a$ is a real number and $m$ and $n$ are integers, then:

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

Applying this rule to our problem, where the base $a$ is 15, and the exponents $m$ and $n$ are 2 and 4 respectively:

• Step 1: Identify the base and exponents: $15^2$ and $15^4$ have the same base.
• Step 2: Add the exponents: $2 + 4 = 6$.
• Step 3: Simplify the expression using the rule: $15^2 \times 15^4 = 15^{2+4} = 15^6$.

Therefore, the simplified expression is $15^6$.

To solve the expression $7^9 \times 7^1$, we need to apply the rules of exponents, specifically the multiplication of powers rule. According to this rule, when we multiply two powers with the same base, we keep the base and add the exponents together.

• The expression can be written as $a^m \times a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
• In this case, our base $a$ is 7, and our exponents are 9 and 1.
• Applying the formula, we have: $7^9 \times 7^1 = 7^{9+1}$.
• Simplifying the exponent: $9 + 1 = 10$.

Thus, the expression simplifies to: $7^{10}$.

All bases are equal and therefore the exponents can be added together.

$8^2\cdot8^3\cdot8^5=8^{10}$

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

Keep in mind that all the terms in the multiplication have the same base, so we will use the previous property:

$2^{10}\cdot2^7\cdot2^6=2^{10+7+6}=2^{23}$Therefore, the correct answer is option c.

According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:$a^n\times a^m=a^{n+m}$

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

$7^{9+1}=7^{10}$

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:

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

We need to simplify the expression $13^3 \times 13^4 \times 13^2$.

To do this, we'll use the multiplication rule for exponents, which states that when multiplying powers with the same base, we add the exponents. Mathematically, $a^m \times a^n = a^{m+n}$. Here, the common base is 13.

Let's apply this rule:

• The given expression is $13^3 \times 13^4 \times 13^2$.
• Using the rule, we add the exponents together: $3 + 4 + 2$.
• Calculate the total exponent: $3 + 4 + 2 = 9$.

Therefore, the simplified expression is $13^9$.

So, the solution to the problem is $13^9$.

To solve this problem, we'll employ the multiplication rule for exponents:

• Step 1: Convert $15$ into exponential form: $15 = 15^1$.
• Step 2: Apply the exponent rule to the expression $15^4 \times 15 \times 15^3$.
• Step 3: Simplify by adding the exponents: $15^4 \times 15^1 \times 15^3 = 15^{4+1+3}$.
• Step 4: Perform the addition: $4 + 1 + 3 = 8$.

Therefore, the simplified form of the expression is $15^8$.

The correct answer matches choice 3, which is: $15^8$.

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 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.