Power of a Power: Variable in the base of the power

We use the power rule of distributing exponents.

$(a^m)^n=a^{m\cdot n}$We apply it in the problem:

$\big((y^6)^8\big)^9=(y^{6\cdot8})^9=y^{6\cdot8\cdot9}=y^{432}$When we use the aforementioned rule twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the last stage we calculate the result of the multiplication in the power exponent.

Therefore, the correct answer is option b.

We use the formula

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

Therefore, we obtain:

$a^{4\times6}=a^{24}$

Let's solve the problem by applying the steps outlined in the analysis.

• Step 1: Identify the expression we need to simplify: $\left(\left(a \times b\right)^3\right)^7$.

• Step 2: Apply the power of a power rule ($\left(x^m\right)^n = x^{m \times n}$) to the entire expression.

Apply the rule:
$\left(\left(a \times b\right)^3\right)^7 = \left(a \times b\right)^{3 \times 7}$ This simplifies to: $\left(a \times b\right)^{21}$

The expression simplifies to $\left(a \times b\right)^{21}$.

Now, let's consider the choices:

• Choice 1: $\left(a \times b\right)^{21}$ is correct, as it matches the result of our simplification.

• Choice 2: $\left(a \times b\right)^{3-7}$ is incorrect, as it incorrectly subtracts the exponents instead of multiplying them.

• Choice 3: $\left(a \times b\right)^{7+3}$ is incorrect, as it incorrectly adds the exponents instead of multiplying them.

• Choice 4: $\left(a \times b\right)^{\frac{7}{3}}$ is incorrect, as it applies division instead of multiplication to the exponents.

Therefore, the correct choice is Choice 1: $\left(a \times b\right)^{21}$.

To solve this problem, we'll apply the power of a power rule for exponents. The rule states that if you have an expression of the form $(x^m)^n$, it simplifies to $x^{m \cdot n}$.

• Step 1: Identify the given expression: $\left(\left(by\right)^8\right)^9$.
• Step 2: Apply the power of a power rule by multiplying the exponents.

Let's work through the solution:
Step 1: We start with $\left(\left(by\right)^8\right)^9$. Here, $\left(by\right)$ is considered as a single base.
Step 2: Apply the power of a power rule: $(by)^{8 \cdot 9}$.
Step 3: Calculate the exponent multiplication: $8 \times 9 = 72$.

Therefore, the simplified expression is $(b \times y)^{72}$.

Analyzing the choices provided:

• Choice 1: $(b \times y)^{17}$ - Incorrect because the exponents should multiply to 72.
• Choice 2: $(b \times y)^1$ - Incorrect because it does not reflect the multiplication of exponents.
• Choice 3: $(b \times y)^{\frac{9}{8}}$ - Incorrect, involves incorrect operations on the exponents.
• Choice 4: $(b \times y)^{72}$ - Correct as it correctly applies the power of a power rule.

Thus, the correct answer is Choice 4: $\left(b \times y\right)^{72}$.

To solve this problem, we'll simplify the expression $\left(\left(7\times a\right)^4\right)^3$ using the rules of exponents, specifically the power of a power rule.

Let's follow these steps:

• Step 1: Identify the structure of the expression
• Step 2: Apply the power of a power rule
• Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The expression given is $\left(\left(7 \times a\right)^4\right)^3$. This is an example of a power raised to another power.

Step 2: According to the power of a power rule, $(b^m)^n = b^{m \times n}$, we multiply the exponents. Here, the base is $7 \times a$, the first exponent (m) is 4, and the second exponent (n) is 3.

Step 3: Multiply the exponents:

$m \times n = 4 \times 3 = 12$

Thus, the expression simplifies to $(7 \times a)^{12}$.

Therefore, the solution to the problem is $\left(7 \times a\right)^{12}$.

Finally, let's verify our solution against the provided choices:

• Choice 1: $\left(7\times a\right)^1$ - This is incorrect because the exponents aren't simplified correctly.
• Choice 2: $\left(7\times a\right)^7$ - This is incorrect for the same reason.
• Choice 3: $\left(7\times a\right)^{12}$ - This matches our simplified solution.
• Choice 4: $\left(7\times a\right)^{\frac{3}{4}}$ - This doesn't match because the power of a power rule doesn't lead to fractional exponents in this problem.

Hence, Choice 3 is the correct choice and the answer is $\left(7 \times a\right)^{12}$.

To solve the problem, we need to simplify the expression $\left(\left(b \times 6\right)^5\right)^2$.

We will apply the power of a power rule in exponents, which states:

• For an expression $(x^m)^n$, it simplifies to $x^{m \times n}$.

Applying this rule to our expression:

$\left(\left(b \times 6\right)^5\right)^2 = \left(b \times 6\right)^{5 \times 2}$

Calculating the new exponent:

$5 \times 2 = 10$

Therefore, the simplified expression is:

$\left(b \times 6\right)^{10}$

We will now compare this to the given multiple-choice answers:

• Choice 1: $\left(b\times6\right)^3$ - Incorrect
• Choice 2: $\left(b\times6\right)^{10}$ - Correct
• Choice 3: $\left(b\times6\right)^7$ - Incorrect
• Choice 4: $\left(b\times6\right)^{\frac{2}{5}}$ - Incorrect

In conclusion, the correct answer is $\left(b\times6\right)^{10}$, which matches Choice 2.

To solve this problem, we'll follow a structured approach:

• Step 1: Identify the given expression: $\left(\left(x \times y\right)^6\right)^5$.

• Step 2: Apply the Power of a Power rule for exponents.

• Step 3: Simplify the expression to reach the final answer.

Now, let's work through each step:
Step 1: We observe that the expression is $\left(\left(x \times y\right)^6\right)^5$. Here, $(x \times y)$ is raised to the 6th power, and this whole expression is further raised to the 5th power.
Step 2: Apply the Power of a Power rule. This states that if you have an expression $(a^m)^n$, you can simplify it to $a^{m \times n}$.
Therefore, $\left(\left(x \times y\right)^6\right)^5$ becomes $(x \times y)^{6 \times 5}$.
Step 3: Calculate the product of the exponents: $6 \times 5 = 30$. So the expression simplifies to $(x \times y)^{30}$.

Therefore, the solution to the problem is $\left(x \times y\right)^{30}$.

Next, consider the answer choices provided:

• Choice 1: $\left(x \times y\right)^1$ - Incorrect because $6 \times 5 \neq 1$.

• Choice 2: $\left(x \times y\right)^{\frac{5}{6}}$ - Incorrect because $\frac{5}{6}$ does not represent $6 \times 5$.

• Choice 3: $\left(x \times y\right)^{11}$ - Incorrect because $6 \times 5 = 30$, not 11.

• Choice 4: $\left(x \times y\right)^{30}$ - Correct, because the solution matches our simplified expression.

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.

We use the formula

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

Therefore, we obtain:

$((b^3)^6)^2=(b^{3\times6})^2=(b^{18})^2=b^{18\times2}=b^{36}$

We use the power rule for exponents.

$(a^m)^n=a^{m\cdot n}$We apply it to the problem:

$\big((a^2)^3\big)^{\frac{1}{4}}=(a^{2\cdot3})^{\frac{1}{4}}=a^{2\cdot3\cdot\frac{1}{4}}=a^{\frac{6}{4}}=a^{\frac{3}{2}}$When we use the previously mentioned rule twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the third stage we calculate the result of the multiplication in the exponent. While remembering that multiplying by a fraction is actually doubling the numerator of the fraction and, finally, in the last stage we simplify the fraction we obtained in the exponent.

Now remember that -

$\frac{3}{2}=1\frac{1}{2}=1.5$

Therefore, the correct answer is option a.