Multiplication of Powers: System of equations with no solution

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 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 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 $2^2 \times 2^3$, 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 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, 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 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 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 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 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 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 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 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 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 the given problem, we need to simplify the expression $a^2 \times a^3$ using the rules of exponents.

We use the rule for multiplying powers with the same base, which states:

• If you have $a^m \times a^n$, the result is $a^{m+n}$.

Let's apply this rule to the given expression:

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

Simplifying the exponents, we get:

$a^{2+3} = a^5$

In the context of choosing from the given options, the answer corresponding to the application of the multiplication rule before final simplification is:

$a^{2+3}$

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 this problem, we'll follow these steps:

• Step 1: Identify the common base
• Step 2: Apply the property of exponents
• Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The problem gives us the expression $2^a \times 2^2$. Here, the common base is 2.
Step 2: We'll apply the property of exponents, which states that for the same base, you add the exponents: $b^m \times b^n = b^{m+n}$. In this case, it will be $2^{a+2}$.
Step 3: Rewriting the expression using this rule, we get: $2^a \times 2^2 = 2^{a+2}$.

Therefore, the solution to the problem is $2^{a+2}$.

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 $4^{x+2+a}$.