Power of a Power: Number of terms

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 follow these steps:

• Step 1: Identify the given expression.

• Step 2: Apply the power of a power rule using exponent multiplication.

• Step 3: Confirm the result against the given choices.

Now, let's work through each step:

Step 1: Identify the Given Expression.
The given expression is $\left((15 \times 3)^{10}\right)^{10}$.

Step 2: Apply the Power of a Power Rule.
According to the exponent rule, $(a^m)^n = a^{m \times n}$, we multiply the exponents:

• The base is $15 \times 3$.

• The inner exponent is 10, and the outer exponent is 10.

• So, we apply the rule: $((15 \times 3)^{10})^{10} = (15 \times 3)^{10 \times 10}$.

• This simplifies to $(15 \times 3)^{100}$.

Step 3: Confirm the Result Against the Choices.
The expression simplifies to $(15 \times 3)^{100}$.

The correct choice from the options provided is:

$\left(15\times3\right)^{100}$

To solve this problem, we'll apply the power of a power rule on the expression $((4 \times 3)^3)^6$.

Here's how we proceed:

• Step 1: Identify the expression
  The expression given is $((4 \times 3)^3)^6$.

• Step 2: Apply the Power of a Power Rule
  According to the power of a power rule, $(a^m)^n = a^{m \times n}$.
  Therefore, $((4 \times 3)^3)^6 = (4 \times 3)^{3 \times 6}$.

• Step 3: Calculate the Exponent Product
  Multiply the exponents: $3 \times 6 = 18$.

• Step 4: Simplify the Expression
  Thus, we have $(4 \times 3)^{18}$.

Therefore, the simplified expression is $(4 \times 3)^{18}$.

Comparing with the choices provided:

• Choice 1: $(4 \times 3)^2$ - Incorrect.

• Choice 2: $(4 \times 3)^3$ - Incorrect.

• Choice 3: $(4 \times 3)^{18}$ - Correct.

• Choice 4: $(4 \times 3)^{-3}$ - Incorrect.

Thus, the correct answer is: $(4 \times 3)^{18}$, which corresponds to choice 3

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

• Step 1: Analyze the given expression

• Step 2: Apply the appropriate exponent rule

• Step 3: Simplify to reach the final expression

Now, let's work through each step:

Step 1: Begin with the given expression, $\left(\left(7 \times 6\right)^5\right)^{10}$. Here, the inner expression $(7 \times 6)$ is raised to the fifth power, and this result is raised to the tenth power.

Step 2: Use the power of a power rule, which states that $(a^m)^n = a^{m \times n}$. Applying this rule to our expression, we identify $a = (7 \times 6)$, $m = 5$, and $n = 10$.

Step 3: Substitute these values into the formula:

$(7 \times 6)^{5 \times 10} = (7 \times 6)^{50}$

Therefore, the simplified expression is $(7 \times 6)^{50}$.

Upon comparison with the provided answer choices, choice 1 is correct:

• Choice 1: $(7 \times 6)^{50}$ - Correct, matches our simplified result.

• Choice 2: $(7 \times 6)^5$ - Incorrect, doesn't apply exponent rule.

• Choice 3: $(7 \times 6)^2$ - Incorrect, not relevant to problem scope.

• Choice 4: $(7 \times 6)^{15}$ - Incorrect, wrong application of formula.

Therefore, the final answer is $\left(7 \times 6\right)^{50}$.

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

• Step 1: Identify the base and the exponents in the given expression.
• Step 2: Use the power of a power rule to simplify the expression.
• Step 3: Verify the solution against given answer choices.

Now, let's work through each step:

Step 1: The given expression is $((8 \times 9)^{11})^4$. Here, the base is $8 \times 9$, and the original exponent of the entire base is $11$. There is an outer exponent of $4$.

Step 2: Apply the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
Thus, $((8 \times 9)^{11})^4 = (8 \times 9)^{11 \cdot 4}$.

Step 3: Perform the multiplication of exponents:
$11 \cdot 4 = 44$.
Therefore, $((8 \times 9)^{11})^4 = (8 \times 9)^{44}$.

Therefore, the solution to the problem is $(8 \times 9)^{44}$.

Now let's check the provided answer choices:

• Choice 1: $(8 \times 9)^{15}$ - Incorrect, as the operation is $(11 \times 4)$, not $(11 + 4)$.
• Choice 2: $(8 \times 9)^{44}$ - Correct, since $11 \times 4 = 44$.
• Choice 3: $(8 \times 9)^{\frac{4}{11}}$ - Incorrect, as this result is unrelated to multiplied exponents.
• Choice 4: $(8 \times 9)^7$ - Incorrect, as there is no reason to have a resulting exponent of $7$.

Therefore, the correct choice is Choice 2: $(8 \times 9)^{44}$.

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

• Step 1: Identify the given expression.
• Step 2: Apply the "power of a power" rule for exponents.
• Step 3: Perform the necessary calculations to simplify the expression.

Now, let's work through each step:

Step 1: Identify the expression $\left(\left(10 \times 7\right)^8\right)^6$.
Step 2: Using the "power of a power" theorem, which states $(a^m)^n = a^{m \cdot n}$, we apply this to the expression.
Step 3: Inside our expression, $a = 10 \times 7$, $m = 8$, and $n = 6$. Thus, $\left(\left(10 \times 7\right)^8\right)^6$ becomes $\left(10 \times 7\right)^{8 \times 6}$.

Step 4: Calculate the new exponent: $8 \times 6 = 48$. Thus, the expression simplifies to $\left(10 \times 7\right)^{48}$.

Therefore, the solution to the problem is $\left(10 \times 7\right)^{48}$.

Now, let's consider the provided answer choices:

• Choice 1: $\left(10\times7\right)^2$ - Incorrect because this does not align with our calculation.
• Choice 2: $\left(10\times7\right)^{14}$ - Incorrect because the exponent is not calculated as $8 \times 6$.
• Choice 3: $\left(10\times7\right)^{\frac{2}{4}}$ - Incorrect because this exponent is not what results from $8 \times 6$.
• Choice 4: $\left(10\times7\right)^{48}$ - Correct, as it matches our simplified result.

We conclude that the correct solution is option 4.

We use the formula:

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

$(2^2)^3+(3^3)^4+(9^2)^6=2^{2\times3}+3^{3\times4}+9^{2\times6}=2^6+3^{12}+9^{12}$

To solve this problem, we'll use the power of a power property of exponents. This states that when raising a power to another power, we multiply the exponents.

• Step 1: Apply the power of a power rule to the inner expression $(5^3)^4$.
• Step 2: Simplify the result from Step 1 using the power of a power rule again with the outer power.

Let's break it down step-by-step:

Step 1:
We start with the innermost expression: $(5^3)^4$. According to the power of a power property, $(a^m)^n = a^{m \cdot n}$, so:

$(5^3)^4 = 5^{3 \cdot 4} = 5^{12}$

Step 2:
Now take the result from step 1, $5^{12}$, and raise it to the 6th power:

$(5^{12})^6 = 5^{12 \cdot 6} = 5^{72}$

Therefore, the simplified expression is $\boxed{5^{72}}$.

Matching this result with the choices provided, the correct answer is:

$5^{72}$

Choice (5^{72}) is correct.

When evaluating the incorrect choices:

• Choice (5^{13}): Incorrect because the exponents are not simply added.
• Choice (5^{18}): Incorrect because the exponents are not multiplied accurately.
• Choice (5^{65}): Incorrect as it's derived from a miscalculation.

I am confident that the solution is correct, as it follows directly from applying the correct exponent rules thoroughly and logically.

To solve this problem, we need to simplify the expression $\left(\left(4^2\right)^3\right)^5$ using the rules of exponents.

Let's break down the problem step by step:

• Step 1: Apply the power of a power rule to $(4^2)^3$, which states that $(a^m)^n = a^{m \cdot n}$.
  Thus, $(4^2)^3 = 4^{2 \cdot 3} = 4^6$.
• Step 2: Now apply the power of a power rule again to the result we obtained in Step 1: $(4^6)^5$.
  Using the rule again, this becomes $(4^6)^5 = 4^{6 \cdot 5} = 4^{30}$.

Therefore, the simplified expression is $4^{30}$.

Let's verify against given choices:

• Choice 1: $4^{30}$ - This is correct.
• Choice 2: $4^{25}$ - Incorrect, results from missing a power multiplication.
• Choice 3: $4^{10}$ - Incorrect, results from only one power application.
• Choice 4: $4^{16}$ - Incorrect, possibly confusing the base calculation.

Thus, the correct answer is indeed $4^{30}$.

Let's handle each expression in the problem separately:

a. We'll start with the leftmost expression, first calculating the result of the multiplication in parentheses, and then use the power rule for power to a power:

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

Let's apply this to the problem for the first expression from the left:

$((7\cdot3)^2)^6=(21^2)^6=21^{2\cdot6}=21^{12}$

where in the final step we calculated the result of multiplication in the power expression,

We're done with this expression, let's move on to the next expression from the left.

b. Let's continue with the second expression from the left, using the power rule for power to a power that we mentioned above and apply it separately to each factor in this expression:

$(3^{-1})^3\cdot(2^3)^4=3^{-1\cdot3}\cdot2^{3\cdot4}=3^{-3}\cdot2^{12}$

Note that the multiplication factors we got have different bases, so we cannot further simplify this expression,

Therefore, let's combine parts a and b above in the result of the original problem:

$((7\cdot3)^2)^6+(3^{-1})^3\cdot(2^3)^4=21^{12}+3^{-3}\cdot2^{12}$

Therefore, the correct answer is answer d.

Let's solve this in two stages. In the first stage, we'll use the power rule for powers in parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

which states that when a power is applied to terms in parentheses, it applies to each term inside the parentheses when they are opened,

Let's apply this rule to our problem:

$\big((x^{\frac{1}{4}}\cdot3^2\cdot6^3)^{\frac{1}{4}}\big)^8=((x^{\frac{1}{4}})^{\frac{1}{4}}\cdot(3^2)^{\frac{1}{4}}\cdot(6^3)^{\frac{1}{4}})^8$

where when opening the parentheses, we applied the power to each term separately, but since each of these terms is raised to a power, we did this carefully and used parentheses,

Next, we'll use the power rule for a power raised to a power:

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

Let's apply this rule to the expression we got:

$(x^{\frac{1}{4}\cdot\frac{1}{4}}\cdot3^{2\cdot\frac{1}{4}}\cdot6^{3\cdot\frac{1}{4}})^8=(x^{\frac{1}{16}}\cdot3^{\frac{2}{4}}\cdot6^{\frac{3}{4}})^8=x^{\frac{1}{16}\cdot8}\cdot3^{\frac{2}{4}\cdot8}\cdot6^{\frac{3}{4}\cdot8}=x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}$

where in the second stage we performed multiplication in the fractions of the power expressions of the terms we obtained, remembering that multiplication in fractions is actually multiplication in the numerator, and then - in the final stage we simplified the fractions in the power expressions of the multiplication terms we got:

$x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}=x^{\frac{1}{2}}\cdot3^4\cdot6^6$

Therefore, the correct answer is answer B.

We will solve the problem in two steps, in the first step we will use the power of a product rule:

$(z\cdot t)^n=z^n\cdot t^n$The rule states that the power affecting a product within parentheses applies to each of the elements of the product when the parentheses are opened,

We begin by applying the law to the given problem:

$(y^3\cdot x^2)^4=(y^3)^4\cdot(x^2)^4$When we open the parentheses, we apply the power to each of the terms of the product separately, but since each of these terms is already raised to a power, we must be careful to use parentheses.

We then use the power of a power rule.

$(b^m)^n=b^{m\cdot n}$We apply the rule to the given problem and we should obtain the following result:

$(y^3)^4\cdot(x^2)^4=y^{3\cdot4}\cdot x^{2\cdot4}=y^{12}\cdot x^8$When in the second step we perform the multiplication operation on the power exponents of the obtained terms.

Therefore, the correct answer is option d.

Let's solve this in two stages. In the first stage, we'll use the power law for power of a product in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

which states that when raising a product in parentheses to a power, the power applies to each factor of the product when opening the parentheses,

Let's apply this law to our problem:

$(x^2\cdot y^3\cdot z^4)^2=(x^2)^2\cdot(y^3)^2\cdot(z^4)^2$

where when opening the parentheses, we applied the power to each factor of the product separately, but since each of these factors is raised to a power, we did this carefully and used parentheses,

Next, we'll use the power law for power of a power:

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

Let's apply this law to the expression we got:

$(x^2)^2\cdot(y^3)^2\cdot(z^4)^2=x^{2\cdot2}\cdot y^{3\cdot2}\cdot z^{4\cdot2}=x^4\cdot y^6\cdot z^8$

where in the second stage we performed the multiplication operation in the power exponents of the factors we obtained.

Therefore, the correct answer is answer D.

Note:

From the above formulation of the power law for parentheses, it might seem that it only refers to two factors in a product within parentheses, but in fact, it is valid for a power of a product of multiple factors in parentheses, as demonstrated in this problem and others,

It would be a good exercise to prove that if the above law is valid for a power of a product of two factors in parentheses (as it is formulated above), then it is also valid for a power of a product of multiple factors in parentheses (for example - three factors, etc.).

We'll use the power rule for a power:

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

We'll apply this rule to the expression in the problem in two stages:

$((9xyz)^3)^4+(a^y)^x= (9xyz)^{3\cdot4}+(a^y)^x=(9xyz)^{12}+a^{yx}$

In the first stage, for good order, we applied the above power rule first to the first term in the expression and dealt with the outer parentheses, then we simplified the expression in the exponent while simultaneously applying the mentioned power rule to the second term in the sum in the problem's expression,

We'll continue and recall the power rule for powers that applies to parentheses containing multiplication of terms:

$(w\cdot t)^n=w^n\cdot t^n$

We'll apply this power rule to the expression we got in the last stage:

$(9xyz)^{12}+(a^y)^x =9^{12} x^{12} y^{12}z^{12}+a^{yx}$

When we applied the mentioned power rule to the first term in the sum in the expression we got in the last stage, and applied the power on the parentheses to each of the multiplication terms inside the parentheses,

Let's summarize the solution steps so far, we got that:

$((9xyz)^3)^4+(a^y)^x=(9xyz)^{12}+a^{yx} =9^{12} x^{12} y^{12} z^{12}+a^{yx}$

Therefore the correct answer is answer D.

In solving the problem we will use two laws of exponents, let's recall them:

a. The law of exponents for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

b. The law of exponents for power of a power:

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

We will apply these two laws of exponents to the expression in the problem in two stages:

We'll start by applying the law of exponents mentioned in a to the second term from the left in the expression:

$9^{-3}\cdot9^4=9^{-3+4}=9^1=9$

After we applied the law of exponents mentioned in a in the first stage and simplified the resulting expression,

We'll continue to the next stage and apply the law of exponents mentioned in b and deal with the third term from the left in the expression, we'll do this in two steps:

$((7^2)^5)^6=(7^2)^{5\cdot6}=7^{2\cdot5\cdot6}=7^{60}$

When in the first step we applied the law of exponents mentioned in b and eliminated the outer parentheses, in the next step we applied the same law of exponents again and eliminated the remaining parentheses, in the following steps we simplified the resulting expression,

Let's summarize the two stages detailed above for the complete solution of the problem:

$(9\cdot7\cdot6)^3+9^{-3}\cdot9^4+((7^2)^5)^6+2^4 = (9\cdot7\cdot6)^3+9+7^{60}+2^4$

In the next step we'll calculate the result of multiplying the terms inside the parentheses in the leftmost term:

$(9\cdot7\cdot6)^3+9+7^{60}+2^4 =378^3+9+7^{60}+2^4$

Therefore the correct answer is answer d.