Power of a Power: System of equations with no solution

To solve this problem, we need to simplify the expression $\left(6^2\right)^7$ using the power of a power rule.

The power of a power rule states that when you have an expression of the form $(a^m)^n$, this can be simplified to $a^{m \times n}$.

Let's apply this rule to the given expression:

1. Identify the base and exponents: - Base: $6$ - First exponent (inside parenthesis): $2$ - Second exponent (outside parenthesis): $7$

2. Apply the power of a power rule: - Simplify $(6^2)^7 = 6^{2 \times 7}$.

3. Calculate the final exponent: - Multiply the exponents: $2 \times 7 = 14$. - Therefore, the simplified expression is $6^{14}$.

Considering the answer choices provided:

• Choice 1: $6^{2 \times 7}$ (Correct, as per our solution).
• Choice 2: $6^{2 + 7}$ (Incorrect, addition is used instead of multiplication).
• Choice 3: $6^{7-2}$ (Incorrect, subtraction is used incorrectly).
• Choice 4: $6^{\frac{7}{2}}$ (Incorrect, division is used incorrectly).

Thus, the correct answer to the problem is $6^{2 \times 7}$, which simplifies to $6^{14}$, and aligns with Choice 1.

To solve this problem, let's carefully follow these steps:

• Step 1: Identify the base and exponents in the expression.
• Step 2: Use the power of a power rule to simplify the expression.
• Step 3: Choose the appropriate option from the given answer choices.

Now, let's break this down:

Step 1: The expression given is $(4^5)^2$. Here, the base is 4, the inner exponent is 5, and the outer exponent is 2.

Step 2: We apply the power of a power rule for exponents, which states that $(a^m)^n = a^{m \cdot n}$.

Using the rule, we have:

This means the expression $(4^5)^2$ can be simplified to $4^{10}$.

Step 3: From the answer choices provided, we need to select the one corresponding to $4^{5 \cdot 2}$:

• Choice 1: $4^{\frac{2}{5}}$ - This is incorrect because it deals with division of exponents and not multiplication.
• Choice 2: $4^{5-2}$ - This is incorrect as it incorrectly subtracts the exponents.
• Choice 3: $4^{5 \times 2}$ - This is the correct choice.
• Choice 4: $4^{5+2}$ - This is incorrect as it incorrectly adds the exponents.

Therefore, the solution to the problem is $4^{5 \times 2} = 4^{10}$, which corresponds to choice 3.

To solve this problem, we'll utilize the Power of a Power rule of exponents, which states:

$(a^m)^n = a^{m \cdot n}$

Given the expression $(3^2)^4$, we need to simplify this by applying the rule:

• Step 1: Recognize that we have a base of 3, with an exponent of 2, raised to another exponent of 4.
• Step 2: According to the Power of a Power rule, we multiply the exponents: $2 \times 4$.
• Step 3: Compute the product of the exponents: $2 \times 4 = 8$.
• Step 4: Rewrite the expression as a single power: $3^8$.

This simplifies the original expression $(3^2)^4$ to $3^{8}$.

Comparing this with the given choices:

• Choice 1: $3^{2 \times 4}$ is equivalent to $3^8$, confirming it matches our solution.
• Choices 2, 3, and 4 involve incorrect operations with exponents (addition, subtraction, division) and therefore do not align with the necessary Power of a Power rule.

Thus, the correct answer to the problem is:

$3^{8}$, and this corresponds to Choice 1: $3^{2 \times 4}$.

We are given the expression $(2^2)^3$ and need to simplify it using the laws of exponents and identify the corresponding expression among the choices.

To simplify the expression $(2^2)^3$, we use the "power of a power" rule, which states that $(a^m)^n = a^{m \times n}$.

Applying this rule to our expression, we have:

Calculating the new exponent:

Thus, the expression simplifies to:

Now, let's compare our result $2^6$ with the given choices:

• Choice 1: $2^{2+3} = 2^5$ - Incorrect, as our expression evaluates to $2^6$, not $2^5$.
• Choice 2: $2^{2-3} = 2^{-1}$ - Incorrect, as our expression evaluates to $2^6$, not $2^{-1}$.
• Choice 3: $2^{\frac{2}{3}}$ - Incorrect, as our expression evaluates to $2^6$, not a fractional exponent expression.
• Choice 4: $2^{2 \times 3} = 2^6$ - Correct, as this matches our simplified expression.

Therefore, the correct choice is Choice 4: $2^{2 \times 3}$.

To solve this problem, we will proceed with the following steps:

• Identify the expression structure.
• Apply the power of a power rule for exponents.
• Simplify the expression.

Now, let's work through each step in detail:

Step 1: Identify the expression structure.
We have the expression $(10^3)^3$. This indicates a power of a power where the base is 10, the inner exponent is 3, and the entire expression is raised to another power of 3.

Step 2: Apply the power of a power rule.
The rule states $(a^m)^n = a^{m \times n}$. Applying this to our specific expression gives us:

Step 3: Perform the multiplication in the exponent.
Calculating $3 \times 3$, we get $9$. Thus, the expression simplifies to:

Therefore, the solution to the problem is:

Examining the provided choices:

• Choice 1: $10^{3+3}$ - Incorrect, because it uses addition instead of multiplication.
• Choice 2: $10^{3 \times 3}$ - Correct, as it matches our derived expression.
• Choice 3: $10^{\frac{3}{3}}$ - Incorrect, because it uses division instead of multiplication.
• Choice 4: $10^{3-3}$ - Incorrect, because it uses subtraction instead of multiplication.

The correct answer is $10^{3 \times 3}$, which is represented by Choice 2.

To solve the problem, we will simplify the expression $\left(\left(2 \times 3\right)^2\right)^5$ using the power of a power exponent rule. Follow these steps:

• Step 1: Identify the form of the expression. The given expression is $\left(\left(2 \times 3\right)^2\right)^5$.
• Step 2: Apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$.
• Step 3: Here, the base is $2 \times 3$, the first exponent ($m$) is 2, and the second exponent ($n$) is 5.
• Step 4: Multiply the exponents: $2 \times 5 = 10$.

Therefore, the expression simplifies to $\left(2 \times 3\right)^{10}$. However, for the purpose of matching the form requested, it can be expressed as $\left(2 \times 3\right)^{2 \times 5}$.

Next, we evaluate the given choices:

• Choice 1: $\left(2 \times 3\right)^{2+5}$ — This incorrectly adds the exponents instead of multiplying them.
• Choice 2: $\left(2 \times 3\right)^{5-2}$ — This incorrectly subtracts the exponents.
• Choice 3: $\left(2 \times 3\right)^{2\times5}$ — This correctly multiplies the exponents, which we found is the right simplification.
• Choice 4: $\left(2 \times 3\right)^{\frac{5}{2}}$ — This introduces division of exponents, which is not applicable here.

The correct choice is Choice 3: $\left(2 \times 3\right)^{2\times5}$.

To solve this problem, we'll use the power of a power property of exponents, which states that for any base $a$ and exponents $m$ and $n$, $(a^m)^n = a^{m \times n}$.

• Step 1: Identify the base and exponents:
  In the given expression $\left(\left(4 \times 6\right)^3\right)^4$, the base is $(4 \times 6)$, the inner exponent is 3, and the outer exponent is 4.

• Step 2: Apply the power of a power rule:
  According to the rule, $\left((4 \times 6)^3\right)^4$ simplifies to $(4 \times 6)^{3 \times 4}$.

• Step 3: Calculate the new exponent:
  Multiply the exponents: $3 \times 4 = 12$. Hence, the expression simplifies to $(4 \times 6)^{12}$.

The expression $\left(\left(4 \times 6\right)^3\right)^4$ is equivalent to $(4 \times 6)^{3 \times 4}$. Therefore, the correct choice is:

$\left(4\times6\right)^{3\times4}$

Therefore, the correct answer is Choice 1.

To solve the problem, we need to simplify the expression $\left(\left(3\times8\right)^5\right)^6$.

We will utilize the "power of a power" rule in exponents, which states $(a^m)^n = a^{m \times n}$. This rule tells us to multiply the exponents when raising a power to another power.

• Step 1: Identify the expression to simplify: $\left(\left(3 \times 8\right)^5\right)^6$.
• Step 2: Apply the power of a power rule: This gives us $(3 \times 8)^{5 \times 6}$.
• Step 3: Multiply the exponents: $5 \times 6 = 30$.

Therefore, the expression simplifies to $(3 \times 8)^{30}$.

Upon comparing this result with the provided answer choices, the correct choice is:

$\left(3\times8\right)^{5\times6}$

This choice correctly applies the power of a power rule, thereby validating the solution as correct.

In conclusion, the simplified form of the expression is $(3 \times 8)^{30}$, and the correct choice is option 4.

To solve this problem, we'll simplify the expression $(\left(10 \times 2\right)^7)^3$ using the rules of exponents:

• Step 1: Identify the structure of the expression
• Step 2: Apply the power of a power rule
• Step 3: Verify the correctness of our answer with the provided choices

Now, let's work through each step:

Step 1: The expression $(\left(10 \times 2\right)^7)^3$ involves two operations: the multiplication inside the parentheses and the power raised to another power outside.

Step 2: We use the power of a power rule $(a^m)^n = a^{m \times n}$. Applying this to the base $\left(10 \times 2\right)$ and the exponents 7 and 3, we have:

$(\left(10 \times 2\right)^7)^3 = \left(10 \times 2\right)^{7 \times 3}$

This simplifies further to:

$\left(10 \times 2\right)^{21}$

Step 3: Now, let's verify with the given choices:
- Choice 1: $\left(10 \times 2\right)^{7+3}$, incorrect because it applies addition instead of multiplication of exponents.
- Choice 2: $\left(10 \times 2\right)^{7 \times 3}$, correct, as it correctly follows the power of a power rule.
- Choice 3: $\left(10 \times 2\right)^{7-3}$, incorrect because it subtracts exponents.
- Choice 4: $\left(10 \times 2\right)^{\frac{3}{7}}$, incorrect because it divides the exponents.

Therefore, the correct choice is Choice 2: $\left(10 \times 2\right)^{7 \times 3}$.

Hence, the simplified expression is $\left(10 \times 2\right)^{21}$.

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

• Step 1: Identify the given expression

• Step 2: Apply the appropriate exponent rule

• Step 3: Simplify the expression

Now, let's work through each step:
Step 1: The problem gives us the expression $((12 \times 5)^4)^8$. Here, the base is $12 \times 5$, and the exponents are $4$ and $8$ respectively.
Step 2: We'll use the Power of a Power Rule, which states $(a^m)^n = a^{m \times n}$. This rule allows us to combine the exponents by multiplying them together.
Step 3: Applying this rule, we rewrite the expression as:
$((12 \times 5)^4)^8 = (12 \times 5)^{4 \times 8}$

Therefore, the simplified expression is $(12 \times 5)^{32}$.

Now, let's consider the choices provided:

• Choice 1: $\left(12 \times 5\right)^{4 \times 8}$ - This matches our simplified expression.

• Choice 2: $\left(12 \times 5\right)^{8-4}$ - Incorrect because it subtracts exponents rather than multiplying them.

• Choice 3: $\left(12 \times 5\right)^{4+8}$ - Incorrect because it adds exponents rather than multiplying them.

• Choice 4: $\left(12 \times 5\right)^{\frac{8}{4}}$ - Incorrect because it divides exponents rather than multiplying them.

Hence, the correct choice is Choice 1: $(12 \times 5)^{32}$.

To solve this problem, we'll apply the power of a power rule for exponents. The problem is to simplify the expression $\left(\left(8\times6\right)^{-7}\right)^{-8}$.

• Step 1: Understand the Power of a Power Rule

The power of a power rule states that $(a^m)^n = a^{m \times n}$.

• Step 2: Apply the Rule

Here, the base of the entire expression is $8 \times 6$, the first exponent is $-7$, and the second exponent is $-8$. According to the rule, we multiply the exponents:

$(8 \times 6)^{-7 \times -8}$

• Step 3: Simplify the Exponent Calculation

Calculate the multiplication of the exponents:

$-7 \times -8 = 56$

This results in the expression:

$(8 \times 6)^{56}$

Considering the given choices, carefully cross-check against our simplified expression:

• Choice 1: $\left(8\times6\right)^{-7-8}$ is incorrect - it uses addition instead of multiplication.

• Choice 2: $\left(8\times6\right)^{\frac{-8}{-7}}$ is incorrect - it uses division instead.

• Choice 3: $\left(8\times6\right)^{-7+8}$ is incorrect - it uses addition as well.

• Choice 4: $\left(8\times6\right)^{-7\times-8}$ is our correct transformation before final simplification.

After calculating, the expression $\left(8\times6\right)^{-7\times-8} = (8 \times 6)^{56}$. The corresponding expression reflects Choice 4 before final simplification.

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

• Step 1: Identify the expression given and simplify the base.
• Step 2: Apply the Power of a Power Rule for exponents.
• Step 3: Match the result with the provided answer choices.

Let's work through each step:

Step 1: The problem gives us the expression $\left(\left(2\times4\right)^{-2}\right)^4$. First, simplify the base: $2 \times 4$ equals 8, so the expression becomes $\left(8^{-2}\right)^4$.

Step 2: Apply the Power of a Power Rule, which states: $(a^m)^n = a^{m \times n}$. Here, $a = 8$, $m = -2$, and $n = 4$. Calculate $m \times n = -2 \times 4 = -8$.

Therefore, $\left(8^{-2}\right)^4$ simplifies to $8^{-8}$, which can be expressed back in terms of the original base $(2 \times 4)$. So we write it as $\left(2 \times 4\right)^{-8}$.

Step 3: Check the given choices:

• Choice 1: $\left(2\times4\right)^{-2+4}$ represents an exponent of 2; incorrect.
• Choice 2: $\left(2\times4\right)^{\frac{4}{-2}}$ simplifies to -2; incorrect.
• Choice 3: $\left(2\times4\right)^{-2-4}$ simplifies to -6; incorrect.
• Choice 4: $\left(2\times4\right)^{-2\times4} = \left(2\times4\right)^{-8}$; correct.

Thus, the correct choice is Choice 4: $\left(2\times4\right)^{-2\times4}$.

To simplify the expression $\left(\left(3\times5\right)^{-3}\right)^{-6}$, we apply exponent rules, specifically the power of a power rule.

Here's a step-by-step solution:

• Step 1: Identify the structure of the expression: we have an outer exponent $-6$ and an inner exponent $-3$ applied to the base $(3 \times 5)$.

• Step 2: Apply the power of a power rule, which states $(a^m)^n = a^{m \cdot n}$. This combines the exponents being multiplied.

• Step 3: Calculate the multiplication of the exponents: $-3 \times -6$. This yields $18$ since multiplying two negative numbers results in a positive number.

• Step 4: Substitute back into the expression: $\left(3 \times 5\right)^{18}$.

Thus, the transformation of the original expression results in the new expression:

$\left(3\times5\right)^{-3\times-6}$

Comparing with the provided choices, we see:

• Choice 1: $\left(3\times5\right)^{-3\times-6}$ - This matches our solution.

• Choices 2, 3, and 4 do not match the derived steps based on multiplying exponents.

Thus, the correct answer is choice 1: $\left(3\times5\right)^{-3\times-6}$.

To solve this problem, we'll simplify the expression $\left(\left(6\times2\right)^4\right)^{-5}$ using exponent rules.

Here's a step-by-step breakdown of the solution:

• Step 1: Identify the form
  The expression is $\left((6 \times 2)^4\right)^{-5}$. This is a case of the power of a power: $(a^m)^n$, which can be rewritten as $a^{m \times n}$.
• Step 2: Apply the power of a power rule
  Apply the rule: $\left((6 \times 2)^4\right)^{-5} = (6 \times 2)^{4 \times (-5)}$.
• Step 3: Simplify the exponent
  Calculate the exponent multiplication: $4 \times (-5) = -20$. Thus, the expression simplifies to $(6 \times 2)^{-20}$.

The resulting expression matches the format of choice 3: $\left(6 \times 2\right)^{4\times-5}$.

Therefore, the correct choice is Choice 3, $\left(6\times2\right)^{4\times-5}$.

To simplify the expression $\left(\left(7\times4\right)^{-6}\right)^5$, follow these steps, checking against the choices provided:

Step 1: Apply the power of a power rule.

• The expression inside the parentheses $7\times4$ acts as a single term $a$.

• Therefore, by applying $(a^m)^n = a^{m \times n}$, we simplify:

$\left(\left(7\times4\right)^{-6}\right)^5 = \left(7\times4\right)^{-6 \times 5}$

Step 2: Multiply the exponents.

• Calculate $-6 \times 5 = -30$.

• Hence, the expression simplifies to $\left(7\times4\right)^{-30}$.

Conclusion:

The correct simplified form of the expression is $\left(7\times4\right)^{-6\times5}$, aligning with choice 2 and your provided correct answer.

To solve this problem, we'll apply the power of a power rule from exponents:

• Step 1: Identify the base and the exponents involved.

• Step 2: Apply the power of a power rule.

• Step 3: Simplify the expression by multiplying the exponents.

Now, let's work through each step:
Step 1: The original expression is $\left(\left(a\times3\right)^2\right)^4$. We recognize the base as $a \times 3$ and see it is first raised to the power of 2, and then the result is raised to the power of 4.
Step 2: We'll use the power of a power property of exponents: $(b^m)^n = b^{m \times n}$. Here, $b$ can be considered as $(a \times 3)$, $m = 2$, and $n = 4$.
Step 3: Applying this property, we have $\left(\left(a\times3\right)^2\right)^4 = \left(a\times3\right)^{2 \times 4}$.
By multiplying the exponents, we get $(a \times 3)^8$, but to match the format requested in the choices, we simply express it as $(a \times 3)^{2 \times 4}$.

Therefore, the correct expression that corresponds to the given power structure is $\left(a\times3\right)^{2\times4}$.

Analyzing the choices provided:

• Choice 1: $\left(a\times3\right)^{4-2}$ applies an incorrect operation of subtraction.

• Choice 2: $\left(a\times3\right)^{\frac{4}{2}}$ incorrectly divides the exponents.

• Choice 3: $\left(a\times3\right)^{2\times4}$ correctly applies the power of a power rule.

• Choice 4: $\left(a\times3\right)^{2+4}$ adds the exponents instead of multiplying.

The correct answer is clearly Choice 3: $\left(a\times3\right)^{2\times4}$.

To solve this problem, we will apply the power of a power rule for exponents, which states that $\left(a^m\right)^n = a^{m \times n}$.

**Step-by-step Solution:**

• Step 1: Identify the given information.

  • The expression is $\left(\left(4\times x\right)^5\right)^3$.

  • The base is $4\times x$, the first exponent is 5, and the second exponent is 3.

• Step 2: Apply the power of a power rule.

  • According to the rule: $\left((4 \times x)^5\right)^3 = (4 \times x)^{5 \times 3}$.

  • Multiply the exponents: $5 \times 3 = 15$.

  • Thus, the expression simplifies to $(4 \times x)^{15}$.

Therefore, the simplified expression is $(4 \times x)^{15}$.

Choice Analysis:

• The correct choice is:

  $\left(4\times x\right)^{5\times3}$

  , which correctly applies the power of a power rule.

• Incorrect choices:

  • : $(4\times x)^{5+3}$ – Incorrect, it adds exponents instead of multiplying.

  • : $(4\times x)^{\frac{5}{3}}$ – Incorrect, it uses division but should multiply exponents.

  • : $(4\times x)^{5-3}$ – Incorrect, it subtracts exponents instead of multiplying.

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

• Identify the given information: We have the expression $(\left(a \times x\right)^7)^2$.
• Apply the appropriate formula: Use the "power of a power" rule for exponents.
• Perform the necessary calculations: Simplify the expression using the rule.

Now, let's work through each step:
Step 1: The expression given is $(\left(a \times x\right)^7)^2$.
Step 2: According to the power of a power rule, $(b^m)^n = b^{m \times n}$. Hence, $(\left(a \times x\right)^7)^2 = (\left(a \times x\right)^{7 \times 2})$.
Step 3: Perform the multiplication in the exponent, which results in $(\left(a \times x\right)^{14})$.

Therefore, the expression $(\left(a \times x\right)^7)^2$ simplifies to $(a \times x)^{14}$.

Checking against the answer choices:

• Choice 1: $\left(a\times x\right)^{7-2}$ simplifies to $\left(a\times x\right)^5$. Incorrect.
• Choice 2: $\left(a\times x\right)^{\frac{2}{7}}$. Incorrect application of exponent rules.
• Choice 3: $\left(a\times x\right)^{7+2} = \left(a\times x\right)^9$. Incorrect.
• Choice 4: $\left(a\times x\right)^{7\times2} = \left(a\times x\right)^{14}$. Correct.

Given all these considerations, the correct choice is Choice 4: $\left(a\times x\right)^{14}$, which corresponds to the correct application of the "power of a power" rule.

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

• Step 1: Identify the mathematical problem and the rule needed.
• Step 2: Apply the power of a power rule to simplify the expression.
• Step 3: Evaluate our simplification against the provided answer choices.

Now, let's work through each step:

Step 1: The problem involves simplifying the expression $\left(\left(x \times b\right)^5\right)^8$, which is a power of a power. Our goal is to simplify it to a single power.

Step 2: According to the power of a power rule, $(a^m)^n = a^{m \times n}$. In this case, the base is $(x \times b)$ raised to the 5th power and that entire expression is further raised to the 8th power.

Applying the power of a power rule gives us:

$\left(\left(x \times b\right)^5\right)^8 = \left(x \times b\right)^{5 \times 8} = \left(x \times b\right)^{40}$

Step 3: Compare this to the provided answer choices:

• Choice 1: $\left(x \times b\right)^{5+8}$ – This is incorrect as it adds the exponents instead of multiplying them.
• Choice 2: $\left(x \times b\right)^{5 \times 8}$ – This is correct as it applies the rule correctly to yield $(x \times b)^{40}$.
• Choice 3: $\left(x \times b\right)^{\frac{8}{5}}$ – This is incorrect and irrelevant to the power of a power rule.
• Choice 4: $\left(x \times b\right)^{5-8}$ – This is incorrect as it subtracts exponents, which is not applicable here.

Therefore, after careful analysis, the solution to the problem is the expression represented by Choice 2: $\left(x \times b\right)^{40}$.

To solve this problem, we'll implement the following steps:

• Step 1: Identify the initial expression that needs simplification: $\left(\left(by\right)^7\right)^6$.
• Step 2: Apply the "power of a power" rule, which states $(x^m)^n = x^{m \times n}$.
• Step 3: Calculate the product of the exponents $7$ and $6$.

Let's explore each step in detail:
Step 1: We begin with the expression $\left(\left(by\right)^7\right)^6$. This indicates $(by)^7$ is raised to another power, 6.

Step 2: According to the power of a power rule, we multiply the exponents:

Step 3: The resulting expression is simplified and expressed as $(b \times y)^{42}$.

Therefore, the solution to the problem is $\left(b \times y\right)^{42}$.

Now, comparing our result with the provided choice options:

• Choice 1: $\left(b\times y\right)^{7+6}$ - Incorrect, because it adds the exponents, which violates the power of a power rule.
• Choice 2: $\left(b\times y\right)^{7\times6}$ - Correct, as it reflects the calculated exponent multiplication: $42$.
• Choice 3: $\left(b\times y\right)^{7-6}$ - Incorrect, because it subtracts the exponents, irrelevant in this context.
• Choice 4: $\left(b\times y\right)^{\frac{6}{7}}$ - Incorrect, as it divides the exponents, not applicable here.

Thus, the correct answer is indeed Choice 2.