Multiplication of Powers: Number of terms

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.

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 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 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 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 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 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 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

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

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

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 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 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 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 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 $7^{-2-3+5}$.

In this case we have 2 different bases, so we will add what can be added, that is, the exponents of $3$

$3^x\cdot2^x\cdot3^{2x}=2^x\cdot3^{3x}$

To solve the problem of simplifying the expression $y^9 \times y^2 \times y^3$, we will follow these steps:

First, recognize that the expression entails powers of the same base $y$, and we can use the rule for multiplying powers with the same base. This rule states that when multiplying like bases, we add the exponents. Mathematically, this can be expressed as:

• Step 1: Identify the base $y$ and the exponents $9$, $2$, and $3$.
• Step 2: Apply the rule for multiplication of powers $y^9 \times y^2 \times y^3 = y^{9+2+3}$.
• Step 3: Calculate the sum of the exponents: $9 + 2 + 3 = 14$.
• Step 4: Simplify the expression to yield the solution, $y^{14}$.

In reviewing the answer choices:

• Choice 1: $y^{9+2+3}$ represents the fully simplified expression, which agrees with our solution.
• Choice 2: $y^{9+2} \times y^3$ and choice 3: $y^9 \times y^{2+3}$ still maintain some intermediate steps of simplification. Yet, both can eventually be simplified further to $y^{14}$.

Therefore, all expressions represent correct approaches or intermediates toward achieving the correct final form. Thus, All answers are correct.

To solve this problem, we'll use the property of exponents for multiplying powers with the same base:

• Step 1: Identify that all terms have the same base, which is $8$. The equation is given as $8^a \times 8^2 \times 8^x$.

• Step 2: Apply the multiplication property of exponents: $b^m \times b^n = b^{m+n}$.

• Step 3: Add the exponents: $(a) + (2) + (x)$ to get the new exponent for the single base.

By applying these steps, we obtain:

$8^{a+2+x}$

This result matches choice 1, confirming that this is the correct simplified expression.

To solve the expression $5^2 \times 5^a \times 5^3$, we will make use of the exponent rule for multiplication, which states that if you multiply powers with the same base, you add the exponents:

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

Let's apply this rule step by step:

• Step 1: Identify the base of the power terms. In this problem, the base is $5$.
• Step 2: Write down all the exponents. The exponents we have are $2$, $a$, and $3$.
• Step 3: Add the exponents together:
  $2 + a + 3$.
• Step 4: Simplify the sum: $2 + a + 3 = 5 + a$.
• Step 5: Express the final result as a single power:
  The expression can be rewritten using the exponent rule as $5^{5+a}$.

Thus, the final simplified expression is $5^{5+a}$.

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

• Step 1: Confirm the given expression is $4^x \times 4^2 \times 4^a$.
• Step 2: Apply the exponent rule for multiplication of powers: if $b^m \times b^n = b^{m+n}$, use this with base 4.
• Step 3: Add the exponents of each term.

Let's work through these steps:

Step 1: The expression we have is $4^x \times 4^2 \times 4^a$.

Step 2: Since all parts of the product have the same base $4$, we can use the rule for multiplying powers: $4^x \times 4^2 \times 4^a = 4^{x+2+a}$.

Step 3: The simplified expression is obtained by adding the exponents: $x + 2 + a$.

Therefore, the expression $4^x \times 4^2 \times 4^a$ simplifies to $4^{x+2+a}$.