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 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 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 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 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 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 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 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 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 this problem, we'll follow these steps:

• Step 1: Identify that both terms have the same base, which is 4.

• Step 2: Use the exponent rule for multiplication of powers with the same base: $a^m \times a^n = a^{m+n}$.

• Step 3: Add the exponents $-2$ and $-4$.

Now, let's work through these steps:

Step 1: We have the expression $4^{-2} \times 4^{-4}$.

Step 2: Applying the exponent rule, we combine the exponents:

$4^{-2} \times 4^{-4} = 4^{-2 + (-4)}$

Therefore, our answer is $4^{-2-4}$, which matches choice 4.

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, $2^{6-3}$.

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

Let's begin by analyzing the given expression: $8^4 \times 8 \times 8^{-1}$.

Each term has the base 8, allowing us to use the exponent rule directly:

• First, recognize the exponents for each term: the first term $8^4$ has an exponent of 4, the second term $8$ is equivalent to $8^1$, and the third term $8^{-1}$ has an exponent of -1.
• Then, apply the formula by adding the exponents: $4 + 1 - 1$.

The resulting expression for the exponent is $8^{4+1-1}$.

Therefore, the corresponding expression to the original product is $8^{4+1-1}$.

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 $t^7 \times t^2$, we follow these steps:

• Step 1: Identify the base and the exponents. Here, the base is $t$, the first exponent is 7, and the second exponent is 2.
• Step 2: Apply the exponent rule for multiplying powers with the same base, which states $a^m \times a^n = a^{m+n}$.
• Step 3: Add the exponents: $7 + 2 = 9$.
• Step 4: Write the simplified expression: $t^9$.

Therefore, after applying the exponent rule, the simplified form of the expression is $t^9$.

The correct choice among the given options is not specifically listed, but the simplification corresponds to $t^{7+2}$ before explicitly adding to get $t^9$.