Power of a Power: Variables in the exponent of the power

To solve this problem, we will simplify the expression $\left(\left(4 \times 6\right)^4\right)^x$ by following these steps:

• Step 1: Identify the base and exponents. The base is $4 \times 6$ and the inner exponent is $4$, which is then raised to $x$.
• Step 2: Apply the power of a power rule $(a^m)^n = a^{m \cdot n}$. We apply this rule to $\left(\left(4 \times 6\right)^4\right)^x$.
• Step 3: Perform the multiplication of exponents. $(4 \times 6)^{4 \cdot x} = (4 \times 6)^{4x}$.

Now, let's work through each of these steps:

Step 1: We are given the compound expression $\left(\left(4 \times 6\right)^4\right)^x$. The base here is $4 \times 6$, the first exponent is 4, and the second exponent is $x$.

Step 2: Using the formula for a power of a power, we have $(a^m)^n = a^{m \cdot n}$. Substitute the values: $(4 \times 6)^4$ is being raised to $x$.

Step 3: Simplify the expression: We then multiply the exponents $4 \times x$ to get $(4 \times 6)^{4x}$.

Thus, the expression simplifies to $(4 \times 6)^{4x}$.

Reviewing the given choices:

• Choice 1: $\left(4 \times 6\right)^{4+x}$ - Incorrect, as it adds the exponents.
• Choice 2: $\left(4 \times 6\right)^{4-x}$ - Incorrect, as it subtracts the exponents.
• Choice 3: $\left(4 \times 6\right)^{4x}$ - Correct, as it correctly applies the power of a power rule.
• Choice 4: $\left(4 \times 6\right)^{\frac{x}{4}}$ - Incorrect, as it incorrectly applies division to the exponents.

Therefore, the correct choice is choice 3: $(4 \times 6)^{4x}$.

To solve this problem, let's simplify the expression $\left(\left(7\times3\right)^y\right)^x$ using the exponent rules. Here are the steps:

• Step 1: Identify the expression: $\left(\left(7\times3\right)^y\right)^x$.
• Step 2: Apply the power of a power rule for exponents: $(a^m)^n = a^{m \cdot n}$.
• Step 3: This rule states that when you raise an exponent to another power, you multiply the exponents. In this case, we have $\left(7\times3\right)^{y \cdot x}$.
• Step 4: The correct simplification is to combine the exponents: $(7\times3)^{yx}$.

After applying these steps, the simplified expression of $\left(\left(7\times3\right)^y\right)^x$ is $(7\times3)^{yx}$.

Comparing the given choices, choice 1: $(7\times3)^{yx}$ is consistent with the principle of exponents that was applied here.

Thus, the correct answer to the problem is $(7\times3)^{yx}$ as it's the only choice accurately representing the simplification according to exponent rules.

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

• Step 1: Identify the given expression.
• Step 2: Apply the power of a power rule.
• Step 3: Simplify the expression to find the correct answer.

Now, let's work through each step:

Step 1: The initial mathematical expression given is $\left(\left(5\times8\right)^a\right)^b$.
Step 2: We apply the power of a power rule, which states that $\left(x^m\right)^n = x^{m \times n}$.
Step 3: Applying this rule, we get:

$\left(\left(5\times8\right)^a\right)^b = \left(5\times8\right)^{a \times b}$

Thus, the simplified expression is $\left(5\times8\right)^{a \times b}$.

Comparing this with the choices given:

• Choice 1: $(5\times8)^{a-b}$ - Incorrect, as it uses subtraction instead of multiplication.
• Choice 2: $(5\times8)^{a+b}$ - Incorrect, as it uses addition instead of multiplication.
• Choice 3: $(5\times8)^{a \times b}$ - Correct, as it uses multiplication, as derived.
• Choice 4: $(5\times8)^{\frac{a}{b}}$ - Incorrect, as it uses division instead of multiplication.

Therefore, the correct answer is $\left(5\times8\right)^{a\times b}$, which corresponds to choice 3.

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

• Step 1: Identify the given information.
• Step 2: Apply the appropriate exponent rule.
• Step 3: Simplify and check the result against choices.

Now, let's work through each step:

Step 1: The given expression is $\left((2 \times 5)^a\right)^3$. Here, the base is $2 \times 5$, which is 10, but we don't need to compute it because we're focusing on exponent rules. The expression can be interpreted as $(10^a)^3$.

Step 2: We use the power of a power rule, which tells us that $(x^m)^n = x^{m \cdot n}$. In our case, $x = (2 \times 5)$, $m = a$, and $n = 3$. Applying the rule, we get: $((2 \times 5)^a)^3 = (2 \times 5)^{a \cdot 3} = (2 \times 5)^{3a}$

Step 3: The simplified expression is $(2 \times 5)^{3a}$. Comparing this with the given choices:

• Choice 1: $(2 \times 5)^{a+3}$ - Incorrect, as it adds exponents rather than multiplying them.
• Choice 2: $(2 \times 5)^{a-3}$ - Incorrect, as it subtracts exponents rather than multiplying them.
• Choice 3: $(2 \times 5)^{\frac{3}{a}}$ - Incorrect, as it divides exponents instead of multiplying them.
• Choice 4: $(2 \times 5)^{3a}$ - Correct, because it correctly applies the power of a power rule.

Therefore, the correct answer to the problem is $(2 \times 5)^{3a}$, which corresponds to choice 4.

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

• Step 1: Identify the given expression and its structure
• Step 2: Apply the power of a power rule of exponents
• Step 3: Simplify the expression and compare it with the given choices

Now, let's work through each step:

Step 1: We are given the expression $\left(\left(6\times9\right)^x\right)^a$. The base of this expression is $6 \times 9$, while the exponents are nested with $x$ in the first power and $a$ in the outer power.

Step 2: According to the power of a power rule, we have:
$\left(b^m\right)^n = b^{m \times n}$
Here, replace $b$ with $(6 \times 9)$, $m$ with $x$, and $n$ with $a$. Therefore:
$\left(\left(6\times9\right)^x\right)^a = \left(6\times9\right)^{x \times a}$

Step 3: Simplify and compare against the available choices:

• Choice 1: $\left(6\times9\right)^{xa}$
• Choice 2: $\left(6\times9\right)^{x+a}$
• Choice 3: $\left(6\times9\right)^{x-a}$
• Choice 4: $\left(6\times9\right)^{\frac{a}{x}}$

The simplified expression $\left(6\times9\right)^{xa}$ matches Choice 1.

Therefore, the correct answer is Choice 1: $\left(6\times9\right)^{xa}$.

Using the law of powers for an exponent raised to another exponent:

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

$(4^x)^y=4^{xy}$Therefore, the correct answer is option a.

We use the power rule for an exponent raised to another exponent:

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

$((3^9)^{4x})^{5y}= (3^9)^{4x\cdot 5y} =3^{9\cdot4x\cdot 5y}=3^{180xy}$In the first step we applied the previously mentioned power rule and removed the outer parentheses. In the next step we applied the power rule once again and removed the remaining parentheses. In the final step we simplified the resulting expression.

Therefore, the correct answer is option b.

Using the power rule for an exponent raised to another exponent:

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

$((14^{3x})^{2y})^{5a}=(14^{3x})^{2y\cdot5a}=14^{3x\cdot2y\cdot5a}=14^{30xya}$In the first step we applied the aforementioned power rule and removed the outer parentheses. In the next step we again applied the power rule and removed the remaining parentheses.

In the final step we simplified the resulting expression,

Therefore, through the rule of substitution (which is applied to the exponent of the power in the obtained expression) it can be concluded that the correct answer is answer D.

We'll use the power rule for powers:

$(a^m)^n=a^{m\cdot n}$We'll apply this rule to the expression in the problem:

$((4x)^{3y})^2= (4x)^{3y\cdot2}=(4x)^{6y}$When in the first stage we applied the mentioned power rule and eliminated the outer parentheses, in the next stage we simplified the resulting expression,

Next, we'll recall the power rule for powers that applies to parentheses containing a product of terms:

$(a\cdot b)^n=a^n\cdot b^n$We'll apply this rule to the expression we got in the last stage:

$(4x)^{6y} =4^{6y}\cdot x^{6y}$When we applied the power to the parentheses to each term of the product inside the parentheses.

Therefore, the correct answer is answer D.