Multiplication of Powers: Inverse formula

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 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.

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 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 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 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 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 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 tackle this problem, we need to expand the expression $7^{2x+7}$ using the properties of exponents:

• Step 1: Identify the structure of the exponent in the form $2x + 7$.
• Step 2: Recognize this as a sum of two components: $2x$ and $7$.
• Step 3: Apply the exponent rule $a^{m+n} = a^m \times a^n$ to separate the expression into two distinct exponent terms.

Now, let's proceed with the solution:

Step 1: Given the exponent is $2x + 7$, break it down into individual components:
$2x + 7 = x + x + 7$.

Step 2: Applying the rule $a^{m+n} = a^m \times a^n$:
$7^{2x+7} = 7^{x+x+7} = 7^x \times 7^{x+7}$.

Therefore, the expanded form of the expression is $7^x \times 7^{x+7}$.

To solve this problem, we will use the properties of exponents:

• Step 1: Recognize that the exponent $10a + 5x$ can be expressed as the sum of two terms, namely $(5a + 5x) + 5a$.
• Step 2: Apply the property $g^{m+n} = g^{m} \times g^{n}$ to the expression $g^{10a + 5x}$.
• Step 3: Expand the expression using the identified split:
  Since $10a + 5x = (5a + 5x) + 5a$, we have:
  $g^{10a + 5x} = g^{(5a + 5x) + 5a} = g^{5a + 5x} \times g^{5a}$.

Thus, the expanded form of the expression is given by $g^{5a+5x} \times g^{5a}$.

The problem asks us to expand the expression $4^{-6}$ using the rules of exponents.

To start, recognize that the negative exponent $-6$ can be split into smaller parts, which can be achieved by breaking it into two equal parts: $-3 + (-3)$. This means we can rewrite $4^{-6}$ as:

$4^{-6} = 4^{-3 + (-3)} = 4^{-3} \times 4^{-3}$

By expressing $4^{-6}$ as a product of two identical terms, $4^{-3} \times 4^{-3}$, we have expanded the original expression correctly according to the rules of exponents. This uses the property of exponents that states $a^{m+n} = a^m \times a^n$.

Thus, the expanded form of $4^{-6}$ is $4^{-3} \times 4^{-3}$.

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

• Step 1: Identify the power that needs expansion: $b^7$.
• Step 2: Use the rule for the multiplication of powers to express this as a product of smaller powers of $b$.
• Step 3: Decompose 7 into a sum of smaller numbers and express the power as a product of smaller terms.

Now, let's apply these steps:

Step 1: We have $b^7$ and need to write it as a product of powers.

Step 2: Recall the rule for multiplication of powers, $b^a \times b^b = b^{a+b}$. We will express 7 as the sum of smaller numbers.

Step 3: Choose smaller exponents that add up to 7. Here, we can use $1 + 2 + 4 = 7$. Therefore, $b^7 = b^1 \times b^2 \times b^4$.

Therefore, the expanded form of the expression is $b^1 \times b^2 \times b^4$.

To expand the expression $t^9$, we will express it as a product of powers. Using the exponent rule, which states that you can multiply powers with the same base by adding their exponents, we need to find powers such that the sum of the exponents equals 9.

Let's consider the expression $t^4 \times t^2 \times t^3$. When we apply the multiplication rule of exponents with the same base $t$, we have:

• $t^9 = t^4 \times t^2 \times t^3$
• Here, $4 + 2 + 3 = 9$, confirming that the sum of the exponents equals the original exponent.

Thus, the expanded form of $t^9$ is indeed $t^4 \times t^2 \times t^3$, which confirms that this is the correct expansion.

Therefore, the solution to the problem is $\boldsymbol{t^4 \times t^2 \times t^3}$.

We are given the expression $9^{-7}$ and need to express it as a product of lower powers of 9 using negative exponents. This requires using the properties of exponents:

• If $a^m \times a^n = a^{m+n}$, then you can decompose a power into smaller factors.

To decompose $-7$, we seek integers whose sum equals $-7$. A possible set of integers is $-4$, $-2$, and $-1$. Therefore, we express $9^{-7}$ as:

$9^{-7} = 9^{-4} \times 9^{-2} \times 9^{-1}$

Looking at the given multiple-choice options:

• Option 1: $9^{-7} \times 9^{-1}$ would sum the exponents to become $-7 + (-1) = -8$, which is incorrect.
• Option 2: $9^{-4} \times 9^{-2} \times 9^{-1}$ correctly adds $-4 + (-2) + (-1) = -7$, matching our requirement.
• Option 3: $9^7 \times 9^{-1}$ simplifies to $7 - 1 = 6$, not $-7$, so it's incorrect.
• Option 4: "A+C are correct" is incorrect as neither A nor C solve the problem correctly.

Option 2 is correct. Therefore, the solution is:

The expression $9^{-7}$ can be written as $9^{-4} \times 9^{-2} \times 9^{-1}$.

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

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 $4^a \times 4^b \times 4^c$.

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 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 $2^{2a} \times 2^a$.