Powers of a Fraction: Applying the formula

To solve the given power expression, we need to apply the formula for powers of a fraction. The expression we are given is:
$\left(\frac{2}{3}\right)^3$

Let's break down the steps:

• When we raise a fraction to a power, we apply the exponent to both the numerator and the denominator separately. This means raising both 2 and 3 to the power of 3.
• Thus, we calculate:
  $2^3 = 8$ and $3^3 = 27$.
• Therefore, $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$.

So, the result of the expression $\left(\frac{2}{3}\right)^3$ is $\frac{8}{27}$.

We use the formula:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

$(\frac{2}{6})^3=(\frac{2}{2\times3})^3$

We simplify:

$(\frac{1}{3})^3=\frac{1^3}{3^3}$

$\frac{1\times1\times1}{3\times3\times3}=\frac{1}{27}$

To solve this problem, we'll compute $\left(\frac{6}{8}\right)^2$ using the rule for powers of a fraction:

• Step 1: Identify the numerator and denominator: $a = 6$ and $b = 8$.
• Step 2: Apply the formula $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ with $n = 2$.
• Step 3: Calculate $a^n = 6^2$.
• Step 4: Calculate $b^n = 8^2$.

Now, let's carry out the calculations:

Step 3: $6^2 = 36$.

Step 4: $8^2 = 64$.

Therefore, $\left(\frac{6}{8}\right)^2 = \frac{36}{64}$.

We compare $\frac{36}{64}$ with the given answer choices:

• Choice 1: $\frac{12}{16}$ is not equal.
• Choice 2: $\frac{6}{64}$ is not equal.
• Choice 3: $\frac{36}{64}$ matches our result.
• Choice 4: $\frac{36}{8}$ is not equal.

Therefore, the correct answer is $\frac{36}{64}$, which matches Choice 3.

To solve $\left(\frac{3}{2}\right)^3$, follow these steps:

• Step 1: Identify the fraction to be raised to a power, $\frac{3}{2}$, and the exponent, 3.
• Step 2: Apply the power to both the numerator and the denominator separately using the exponent rule for fractions.

Let's evaluate:

Raise the numerator 3 to the power of 3:

$3^3 = 3 \times 3 \times 3 = 27$

Raise the denominator 2 to the power of 3:

$2^3 = 2 \times 2 \times 2 = 8$

Combine these results into the fraction $\frac{27}{8}$.

Therefore, $\left(\frac{3}{2}\right)^3 = \frac{27}{8}$.

Given several answer choices, the correct choice is 4: $\frac{27}{8}$.

To solve the problem of squaring the fraction $\left(\frac{4}{7}\right)^2$, we will follow these steps:

• Step 1: Determine the square of the numerator. The numerator is $4$, and $4^2 = 16$.
• Step 2: Determine the square of the denominator. The denominator is $7$, and $7^2 = 49$.
• Step 3: Combine these results to form the fraction $\frac{16}{49}$.

The computation involves squaring both the numerator and the denominator separately. In conclusion, the squared fraction is:

$\frac{16}{49}$

Therefore, the corresponding expression for $\left(\frac{4}{7}\right)^2$ is $\frac{16}{49}$.

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

• Apply the exponentiation rule to the fraction $\frac{2}{3}$.
• Calculate $2^2$ and $3^2$.
• Form the result as a fraction.
• Compare the result to the provided choices and select the correct one.

Now, let's work through each step:

Step 1: The expression given is $\left(\frac{2}{3}\right)^2$.

Step 2: According to the exponentiation rule, we apply the exponent to both the numerator and the denominator:
$\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2}$.

Step 3: Calculate $2^2$ and $3^2$:
$2^2 = 4$, and $3^2 = 9$.

Step 4: Form the resultant fraction:
Thus, $\frac{2^2}{3^2} = \frac{4}{9}$.

Step 5: Finally, compare this result with the given choices:
Our result $\frac{4}{9}$ matches with choice 2.

Therefore, the solution to the problem is $\frac{4}{9}$.

To solve the expression $\left(\frac{1}{3}\right)^3$, we need to apply the rule for exponents of a fraction, which states:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Using this property, we can rewrite the fraction with its exponent as follows:

$\left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3}$

Now, calculate the powers of the numerator and the denominator separately:

• $1^3 = 1$

• $3^3 = 27$

Thus, putting it all together, we have:

$\frac{1^3}{3^3} = \frac{1}{27}$

This shows that raising both the numerator and the denominator of a fraction to a power involves calculating the power of each part separately and then constructing a new fraction.

The solution to the question is: $\frac{1}{27}$

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

• Step 1: Identify the given fraction and power: $\frac{1}{2}$ and the power $2$.
• Step 2: Apply the formula $(\frac{a}{b})^n = \frac{a^n}{b^n}$.
• Step 3: Calculate the numerator as $1^2 = 1$.
• Step 4: Calculate the denominator as $2^2 = 4$.
• Step 5: Combine the results to find $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$.

Now, let's work through each step in detail:
Step 1: The fraction given is $\frac{1}{2}$ and the power is 2.
Step 2: Applying the formula $(\frac{a}{b})^n = \frac{a^n}{b^n}$, we have:
Step 3: Calculate $1^2 = 1$.
Step 4: Calculate $2^2 = 4$.
Step 5: Compile these to get $\frac{1}{4}$.

Therefore, the solution to the problem is $\frac{1}{4}$, which matches choice 2.

To solve this problem, we will evaluate the expression $\left(\frac{4}{5}\right)^3$ following these steps:

• Step 1: Apply the power to the numerator.
• Step 2: Apply the power to the denominator.
• Step 3: Calculate the results for both and simplify if possible.

Let's go through each step:

Step 1: Calculate the cube of the numerator, which is 4. That is, $4^3 = 4 \times 4 \times 4 = 64$.

Step 2: Calculate the cube of the denominator, which is 5. That is, $5^3 = 5 \times 5 \times 5 = 125$.

Step 3: Combine these results into a fraction: $\frac{64}{125}$.

Therefore, the expression $\left(\frac{4}{5}\right)^3$ simplifies to $\frac{64}{125}$.

Upon comparing with the given choices, the correct answer is choice 1: $\frac{64}{125}$.

To solve the problem of finding the value of $\left(\frac{4}{5}\right)^2$, we will follow these steps:

• Step 1: Identify the numerator and the denominator of the fraction. Here, the numerator is 4, and the denominator is 5.
• Step 2: Apply the exponent to both the numerator and the denominator separately: $\left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2}$.
• Step 3: Calculate $\frac{4^2}{5^2}$. This gives:
  • The square of the numerator is $4^2 = 16$.
  • The square of the denominator is $5^2 = 25$.
  Thus, $\frac{4^2}{5^2} = \frac{16}{25}$.

Since we successfully calculated $\left(\frac{4}{5}\right)^2 = \frac{16}{25}$, this matches the choice labeled $\frac{16}{25}$.

Therefore, the correct expression is $\frac{16}{25}$.

To compute $(\frac{3}{4})^3$, you must raise both the numerator and the denominator to the power of 3:

Numerator: $3^3 = 3 \times 3 \times 3 = 27$

Denominator: $4^3 = 4 \times 4 \times 4 = 64$

Therefore, $(\frac{3}{4})^3 = \frac{27}{64}$.

To evaluate $(\frac{5}{6})^2$, raise both the numerator and the denominator to the power of 2:

Numerator: $5^2 = 5 \times 5 = 25$

Denominator: $6^2 = 6 \times 6 = 36$

Therefore, $(\frac{5}{6})^2 = \frac{25}{36}$.

To determine $(\frac{7}{5})^2$, raise both the numerator and the denominator to the power of 2:

Numerator: $7^2 = 7 \times 7 = 49$

Denominator: $5^2 = 5 \times 5 = 25$

Therefore, $(\frac{7}{5})^2 = \frac{49}{25}$.

To solve $(\frac{3}{4})^2$, you need to square both the numerator and the denominator separately:

1. Square the numerator: $3^2 = 9$

2. Square the denominator: $4^2 = 16$

3. Combine the results to get $\frac{9}{16}$

To solve $(\frac{5}{6})^2$, you need to square both the numerator and the denominator separately:

1. Square the numerator: $5^2 = 25$

2. Square the denominator: $6^2 = 36$

3. Combine the results to get $\frac{25}{36}$

To solve the problem $300^{-4} \cdot \left(\frac{1}{300}\right)^{-4}$, let's follow these steps:

• Step 1: Simplify the reciprocal power expression.
  The expression $\left(\frac{1}{300}\right)^{-4}$ can be simplified using the rule for reciprocals, which states $\left(\frac{1}{a}\right)^{-n} = a^n$. Thus, $\left(\frac{1}{300}\right)^{-4} = 300^4$.
• Step 2: Combine the powers.
  Now the expression becomes $300^{-4} \cdot 300^4$. Using the rule $a^m \cdot a^n = a^{m+n}$, we have $300^{-4+4} = 300^0$.
• Step 3: Calculate the final result.
  By the identity $a^0 = 1$, any non-zero number raised to the power of zero equals 1. Therefore, $300^0 = 1$.

Thus, the solution to the problem is $\boxed{1}$.

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

• Step 1: Express the given expression $\left(\frac{a \times b}{2 \times x}\right)^3$ using exponent rules for fractions.
• Step 2: Apply the cube power to both the numerator and the denominator separately.
• Step 3: Simplify the expression and compare it to the provided choices.

Now, let's work through each step:

Step 1: We have the expression $\left(\frac{a \times b}{2 \times x}\right)^3$. According to the power of a fraction rule, $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$, we can raise the numerator and denominator to the power of 3 separately.

Step 2: Apply the cube power:

• Numerator: $(a \times b)^3 = a^3 \times b^3$
• Denominator: $(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3$

Step 3: Combine these results:

The expression simplifies to $\frac{a^3 \times b^3}{8 \times x^3}$.

Comparing it to the given choices, we find that this matches Choice 1: $\frac{a^3 \times b^3}{8 \times x^3}$.

Therefore, the solution to the problem is $\frac{a^3 \times b^3}{8 \times x^3}$.

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

• Step 1: Identify the given expression and its structure.
• Step 2: Apply the rule for powers of a fraction, $\left(\frac{p}{q}\right)^n$.
• Step 3: Calculate powers of individual components within the fraction.
• Step 4: Simplify the resulting expression.

Now, let's work through each step:

Step 1: Our given expression is $\left(\frac{a \times 3}{2 \times x}\right)^3$.

Step 2: Apply the power to the entire fraction using $\left(\frac{p}{q}\right)^n = \frac{p^n}{q^n}$, we get:
$\left(\frac{a \times 3}{2 \times x}\right)^3 = \frac{(a \times 3)^3}{(2 \times x)^3}$.

Step 3: Simplify the numerator and the denominator separately:
Numerator: $(a \times 3)^3 = a^3 \times 3^3 = a^3 \times 27$.
Denominator: $(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3$.

Step 4: Combine the simplified components to form the final expression:
The expression is $\frac{a^3 \times 27}{8 \times x^3}$.

Therefore, the solution to the problem is $\frac{a^3 \times 27}{8 \times x^3}$, which corresponds to choice 2.

To solve this problem, we'll apply the rule for powers of a fraction:

• Step 1: Identify the fraction. The fraction given is $\frac{6}{x \times y}$.
• Step 2: Apply the power to both the numerator and the denominator. This means squaring both 6 and $x \times y$.
• Step 3: Calculate the square of the numerator and the denominator:
• The square of the numerator: $6^2 = 36$.
• The square of the denominator: $(x \times y)^2 = x^2 \times y^2$.
• Step 4: Combine the results: $\left(\frac{6}{x \times y}\right)^2 = \frac{6^2}{(x \times y)^2} = \frac{36}{x^2 \times y^2}$.

Thus, the expression $\left(\frac{6}{x \times y}\right)^2$ simplifies to $\frac{36}{(x \times y)^2}$.

Therefore, the correct answer is clearly the expression $\frac{36}{(x \times y)^2}$, which matches choice 4.

The task is to simplify $\left(\frac{2\times a}{3}\right)^2$.

First, applying the exponent rule for fractions, $\left(\frac{b}{c}\right)^n = \frac{b^n}{c^n}$, we have:

• $\left(\frac{2\times a}{3}\right)^2 = \frac{(2 \times a)^2}{3^2}$ - which represents performing the exponentiation separately on the numerator and denominator.

Now, simplify each part:

• The numerator: $(2 \times a)^2 = 2^2 \times a^2 = 4 \times a^2$.
• The denominator: $3^2 = 9$.

Thus, the expression simplifies to $\frac{4 \times a^2}{9}$.

We ensure the valid transformations based on the choices provided:

• Choice 1: $\frac{4\times a^2}{9}$, which matches what we calculated.
• Choice 2: $\frac{2^2\times a^2}{3^2}$, which is an equivalent form before final multiplication.
• Choice 3: $\frac{\left(2\times a\right)^2}{3^2}$, presenting the step before breaking down the $(2a)^2$.
• Choice 4: "All answers are correct", recognizing all transformations as valid.

Therefore, each choice represents correct steps or forms towards the simplified expression.

The correct answer is: All answers are correct.