Multiplication of Powers: Inverse formula

Let's solve this problem step-by-step using the rules of exponents:

• Step 1: Identify the expression. We are given $6^{3+2}$.

• Step 2: Apply the exponent rule $a^{m+n} = a^m \times a^n$. This allows us to split the addition in the exponent into separate multiplicative terms.

• Step 3: Break down the exponent addition $3+2$ into: $6^3 \times 6^2$.

By applying the rules of exponents, the expression $6^{3+2}$ can be expanded to:
$6^3 \times 6^2$

Therefore, the expanded form of the expression is $6^3 \times 6^2$.

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

• Step 1: Identify the base and the exponents in the expression
• Step 2: Apply the exponent rule $a^{m+n} = a^m \times a^n$
• Step 3: Rewrite the expression using the rule

Now, let's work through each step:
Step 1: The problem gives us the expression $(4)^{4+6}$. Here, the base is 4, and the exponent is the sum $4 + 6$.
Step 2: We'll apply the rule $a^{m+n} = a^m \times a^n$, which allows us to write the expression as the product of two powers.
Step 3: According to the rule, $(4)^{4+6}$ becomes $(4)^4 \times (4)^6$.

This means that $(4)^{4+6}$ expands to $(4)^4 \times (4)^6$.

Therefore, the solution to the problem is $\boxed{4^4 \times 4^6}$, corresponding to choice 4.

To solve this problem, we'll apply the rule for adding exponentials:

• Step 1: Identify the base and the exponents.
  The base is $2$ and the exponents, when added, are $2 + 5$.
• Step 2: Apply the rule for multiplication of powers.
  Using $a^{m+n} = a^m \times a^n$, we have $2^{2+5} = 2^2 \times 2^5$.
• Step 3: Simplify and expand the expression.
  Split the expression into $2^2$ and $2^5$, which is the expanded form based on the power rule.

Therefore, the expanded form of the equation is $2^2 \times 2^5$.

To address this problem, we need to verify the set of exponent rules for each choice provided and determine which, if any, results in $7^8$.

Let's explore and verify the provided choices:

• Choice 1: $7^2 \times 7^4$

Using the rule $a^m \times a^n = a^{m+n}$, we have:

$7^2 \times 7^4 = 7^{2+4} = 7^6$

This does not equal $7^8$.

• Choice 2: $7^8 \times 7^1$

Using the rule $a^m \times a^n = a^{m+n}$, we have:

$7^8 \times 7^1 = 7^{8+1} = 7^9$

This does not equal $7^8$.

• Choice 3: $7^4 \times 7^2$

Using the rule $a^m \times a^n = a^{m+n}$, we have:

$7^4 \times 7^2 = 7^{4+2} = 7^6$

This does not equal $7^8$.

Upon evaluating all given choices, none of the expressions equal $7^8$.

Therefore, based on the analysis, the correct choice is None of the answers are correct.

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

• Step 1: Understand the given information, which is the expression $8^4$.
• Step 2: Check each choice to see if it is a valid decomposition of $8^4$.
• Step 3: Validate each decomposition by applying the formula $a^m \times a^n = a^{m+n}$.

Now, let's work through each step:
Step 1: The given expression is $8^4$. We need to expand it using the properties of exponents.
Step 2: Check each choice:
- Choice 1: $8^1 \times 8^3$. According to the law $a^m \times a^n = a^{m+n}$, this gives $8^{1+3} = 8^4$. Correct.
- Choice 2: $8^2 \times 8^2$. Similarly, $8^{2+2} = 8^4$. Correct.
- Choice 3: $8^3 \times 8^1$. This gives $8^{3+1} = 8^4$. Correct.
Step 3: All choices decompose $8^4$ correctly into powers of 8 that multiply back to the original value, confirming their validity.

Therefore, the solution to the problem is all answers are correct.

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 $6^2 \times 6^3 \times 6^7$.

Let's solve the problem step by step:

The expression given is $10^{-1}$. A negative exponent indicates a reciprocal, so:

$10^{-1} = \frac{1}{10}$

We can express this as a multiplication form of powers of 10:

Using the property of exponents, specifically the multiplication of powers, we can rewrite:

$\frac{1}{10} = 10^{-11} \times 10^{10}$

To verify:

• Apply the rule of exponents: $10^{-11} \times 10^{10} = 10^{-11 + 10} = 10^{-1}$

• This confirms the expression is correctly transformed back to $10^{-1}$.

Thus, the expanded expression of $10^{-1}$ is:

$10^{-11}\times10^{10}$

To solve this problem, we begin by rewriting the expression that incorporates exponent rules. The expression given is $a^{3+5}$. According to the rule of exponents, when you have a base raised to a power that is a sum, $a^{m+n} = a^m \times a^n$.

Let's apply this rule:

• Write the exponent as a sum: $3 + 5$.
• Apply the exponent rule: $a^{3+5}$ becomes $a^3 \times a^5$.

Thus, the expanded form of $a^{3+5}$ using the rule of exponents is $a^3 \times a^5$.

Finally, comparing with the provided options, choice 1 ( $a^3 \times a^5$) is the correct one, as it correctly uses the exponent rule.

Therefore, the solution to the problem is $a^3\times a^5$.

To solve this problem, we will use the rule of exponents that allows us to expand the sum $a + b + c$ in the exponent:

• Given: $4^{a+b+c}$
• According to the exponent rule $x^{m+n+p} = x^m \times x^n \times x^p$, we can express:
• Step: Break down $4^{a+b+c}$ to:
• $4^a \times 4^b \times 4^c$

Therefore, the expanded form of the equation is $4^a \times 4^b \times 4^c$.

To solve the problem, we can follow these steps:

• Step 1: Recognize that the given expression is $2^{2a+a}$.
• Step 2: Use the Power of a Power Rule for exponents, which allows us to write $a^{m+n} = a^m \times a^n$.
• Step 3: Rewrite the expression as follows:

Given: $2^{2a+a}$

Step 4: Simplify the exponent by splitting it:

Since the expression in the exponent is $2a+a$, we can write:

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

Thus, applying the properties of exponents correctly leads us to the expanded form.

Therefore, the expanded equation is $2^{2a} \times 2^a$.

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

• Step 1: Analyze the given expression's exponent.
• Step 2: Apply the exponent addition rule to expand the expression.
• Step 3: Identify the correct choice from a set of given options.

Now, let's work through each step:
Step 1: The expression given is $3^{2a + x + a}$. Here the exponent is $2a + x + a$.
Step 2: We apply the rule $b^{m+n} = b^m \times b^n$ by rewriting the exponent sum as individual terms: $(2a)$, $x$, and $a$.
Thus, we can rewrite the expression using the property of exponents: $3^{2a + x + a} = 3^{2a} \times 3^x \times 3^a$.
Step 3: Upon expanding, the solution corresponds to option :

$3^{2a}\times3^x\times3^a$

Therefore, the expanded expression is $3^{2a}\times3^x\times3^a$.