Multiplication of Powers: Number of terms

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.

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 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 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 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.

To simplify the expression $6^2 \times 6^5 \times 6$, we apply the rules of exponents because all terms have the same base.

• Identify each power: $6^2$, $6^5$, and $6$. Remember that $6$ is equivalent to $6^1$.

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

• Combine the exponents: $6^{2+5+1}$.

• Calculate the sum of the exponents: $2 + 5 + 1 = 8$.

Therefore, the solution to the problem is $\boxed{6^8}$.

To expand the equation $3^{12+10+5}$, we will apply the rule of exponents that states: when you multiply powers with the same base, you can add the exponents. However, in this case, we are starting with a single term and want to represent it as a product of terms with the base being raised to each of the individual exponents given in the sum. Here’s a step-by-step explanation:

1. Start with the expression: $3^{12+10+5}$.

2. Recognize that the exponents are added together. According to the property of exponents (Multiplication of Powers), we can express a single power with summed exponents as a product of powers:

3. Break down the exponents: $3^{12+10+5} = 3^{12} \times 3^{10} \times 3^5$.

4. As seen from the explanation: $3^{12+10+5}$ is expanded to the product $3^{12} \times 3^{10} \times 3^5$ by expressing each part of the sum as an exponent with the base 3.

The final expanded form is therefore: $3^{12} \times 3^{10} \times 3^5$.

To solve this problem, let's examine the possible answer choices to determine which ones equal $7^6$.

• **Choice 1:** $7^1 \times 7^2 \times 7^4$
  By exponent rules: $7^1 \cdot 7^2 \cdot 7^4 = 7^{1+2+4} = 7^7$.
• **Choice 2:** $7^1 \times 7 \times 7^4$
  Here, $7 = 7^1$. So, $7^1 \cdot 7^1 \cdot 7^4 = 7^{1+1+4} = 7^6$.
• **Choice 3:** $7^2 \times 7^2 \times 7^2$
  Using the rule: $7^2 \cdot 7^2 \cdot 7^2 = 7^{2+2+2} = 7^6$.
• **Choice 4:** This states choices 'b + c are correct'.

After calculations, choices 2 and 3 simplify to $7^6$. Therefore, the correct answer is indeed that choices 'b+c are correct'. Thus, the correct choice is:

Choice 4: b+c are correct

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

• Step 1: Identify the base and exponent given in the expression
• Step 2: Choose an appropriate combination of exponents for the expansion
• Step 3: Verify the selected combination by checking it matches the rule $a^x \times a^y \times a^z = a^{10}$

Now, let's work through each step:
Step 1: We start with the expression $8^{10}$. Our goal is to express this as a product of three powers of 8 that sum to the same exponent.
Step 2: Using the exponent addition rule, we need to find three exponents $a, b,$ and $c$ such that $8^a \times 8^b \times 8^c = 8^{10}$. One possible approach is to try combinations that could plausibly sum to 10. For example, let’s choose $a = 3$, $b = 3$, $c = 4$. Observing that $3 + 3 + 4 = 10$, a valid distribution can be $8^3 \times 8^3 \times 8^4$.
Step 3: Verify if this aligns with the multiplication of powers: $8^3 \times 8^3 \times 8^4 = 8^{3+3+4} = 8^{10}$, confirming that this product is indeed equivalent to $8^{10}$.

Therefore, the correct expanded form of $8^{10}$ is $8^3\times8^3\times8^4$, corresponding to answer choice 2.

To solve this problem, let's expand $6^{12}$ as a product of three powers of 6:

• Step 1: Understand that we need three exponents, $a$, $b$, and $c$, such that $a + b + c = 12$.
• Step 2: Check each combination in the choices:
• Choice 1: $6^3 \times 6^2 \times 6^2 \Rightarrow 3 + 2 + 2 = 7$ (not equal to 12)
• Choice 2: $6^4 \times 6^4 \times 6^3 \Rightarrow 4 + 4 + 3 = 11$ (not equal to 12)
• Choice 3: $6^2 \times 6^3 \times 6^7 \Rightarrow 2 + 3 + 7 = 12$ (equal to 12) ✔
• Choice 4: $6^1 \times 6^{11} \times 6 \Rightarrow 1 + 11 + 1 = 13$ (not equal to 12)
• Step 3: Verify that choice 3, $6^2 \times 6^3 \times 6^7$, correctly expands to $6^{12}$ since $6^2 \times 6^3 \times 6^7 = 6^{2+3+7} = 6^{12}$.

Therefore, the correct expansion of $6^{12}$ is $6^2 \times 6^3 \times 6^7$.

To reduce the expression $2^3 \times 2^4 \times 2^6 \times 2^5$, we apply the rule of multiplication for exponents with the same base, which states that:

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

Following this rule, we add up all the exponents together since they all have the same base, 2:

$3 + 4 + 6 + 5 = 18$.

So, the expression reduces to $2^{18}$.

Thus, the answer is $2^{18}$.

To solve this problem, we will apply the rule for multiplying powers with the same base. This rule states that when multiplying like bases, we can add the exponents. Let's break down the steps:

• Step 1: Identify the exponents of each power of $6$. They are $3$, $5$, $6$, and $4$.
• Step 2: Add these exponents together. This gives us $3 + 5 + 6 + 4$.
• Step 3: Calculate the sum: $3 + 5 + 6 + 4 = 18$.
• Step 4: Write the result as a single power: $6^{18}$.

Thus, by applying the rule for the multiplication of powers, the entire expression simplifies to $6^{18}$.

The correct answer is therefore $6^{18}$, which corresponds with the choice labeled "1" in the given options.

Proceed to solve this in two stages. In the first stage, we'll use the power rule for powers in parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

which states that when a power is applied to terms in parentheses, it applies to each term inside the parentheses when they are opened,

Let's apply this rule to our problem:

$\big((x^{\frac{1}{4}}\cdot3^2\cdot6^3)^{\frac{1}{4}}\big)^8=((x^{\frac{1}{4}})^{\frac{1}{4}}\cdot(3^2)^{\frac{1}{4}}\cdot(6^3)^{\frac{1}{4}})^8$

When opening the parentheses, we applied the power to each term separately, however given that each of these terms is raised to a power, we did this carefully and used parentheses,

Next, we'll use the power rule for a power raised to a power:

$(b^m)^n=b^{m\cdot n}$

Let's apply this rule to the expression that we obtained:

$(x^{\frac{1}{4}\cdot\frac{1}{4}}\cdot3^{2\cdot\frac{1}{4}}\cdot6^{3\cdot\frac{1}{4}})^8=(x^{\frac{1}{16}}\cdot3^{\frac{2}{4}}\cdot6^{\frac{3}{4}})^8=x^{\frac{1}{16}\cdot8}\cdot3^{\frac{2}{4}\cdot8}\cdot6^{\frac{3}{4}\cdot8}=x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}$

In the second stage we performed multiplication in the fractions of the power expressions of the terms that we obtained. Remember that multiplication in fractions is actually multiplication in the numerator. In the final stage we simplified the fractions in the power expressions of the multiplication terms that we obtained:

$x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}=x^{\frac{1}{2}}\cdot3^4\cdot6^6$

Therefore, the correct answer is answer B.