Powers of a Fraction: Applying the formula

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: $\frac{1}{27}$

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 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 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 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 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 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 $\frac{8}{27}$.

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.

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.

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.

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 $\boxed{1}$.

$(\frac{4^2}{7^4})^2=\frac{4^{2\times2}}{7^{4\times2}}=\frac{4^4}{7^8}$

First, let's note that the first term in the multiplication in the problem has an exponent of 0, and any number (different from zero) raised to the power of zero equals 1, meaning:

$X^0=1$Therefore, we get that the expression in the problem is:

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}= 1\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}=(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}$

Later, we will use the law of exponents for negative exponents:

$a^{-n}=\frac{1}{a^n}$

And before we proceed to solve the problem let's understand this law in a slightly different, indirect way:

Let's note that if we treat this law as an equation (which it indeed is in every way), and multiply both sides of the equation by the common denominator which is:

$a^n$we get:

$a^{-n}=\frac{1}{a^n}\\ \frac{a^n}{1} =\frac{1}{a^n}\hspace{8pt} \text{/}\cdot a^n\\ a^n\cdot a^{-n}=1$

Where in the first stage we remembered that any number can be represented as itself divided by 1, we applied this to the left side of the equation, then we multiplied by the common denominator and to know by how much we multiplied each numerator (after reduction with the common denominator) we addressed the question "by how much did we multiply the current denominator to get the common denominator?".

Let's look at the result we got:

$a^n\cdot a^{-n}=1$

Meaning that $a^n,\hspace{4pt}a^{-n}$are reciprocal numbers, or in other words:

$a^n$ is reciprocal to $a^{-n}$ (and vice versa),

and specifically:

$a,\hspace{4pt}a^{-1}$ are reciprocal to each other,

We can apply this understanding to the problem if we also remember that the reciprocal of a fraction is obtained by switching the numerator and denominator, meaning that the fractions:

$\frac{a}{b},\hspace{4pt}\frac{b}{a}$ are reciprocal fractions - which can be easily understood, since their multiplication will clearly give the result 1,

And if we combine this with the previous understanding, we can easily conclude that:

$\big(\frac{a}{b}\big)^{-1}=\frac{b}{a}$

In other words, raising a fraction to the power of negative one will give a result that is the reciprocal fraction, obtained by switching the numerator and denominator.

Let's return to the problem, the expression we got in the last stage is:

$(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}$

We'll use what was explained earlier and note that the fraction in parentheses in the second term of the multiplication is the reciprocal fraction to the fraction in parentheses in the first term of the multiplication, meaning that:$\frac{13}{2}= \big(\frac{2}{13} \big)^{-1}$Therefore we can simply calculate the expression we got in the last stage by converting to a common base using the above understanding:

$(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5} = (\frac{2}{13})^{-2}\cdot\big( (\frac{2}{13})^{-1}\big)^{-5}$

Where we actually just replaced the fraction in parentheses in the second term of the multiplication with its reciprocal raised to the power of negative one as mentioned earlier,

Next we'll recall the law of exponents for power of a power:

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

And we'll apply this law to the expression we got in the last stage:

$(\frac{2}{13})^{-2}\cdot\big( (\frac{2}{13})^{-1}\big)^{-5} =(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{(-1)\cdot(-5)}=(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5}$

Where in the first stage we applied the above law of exponents and then simplified the expression that resulted,

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

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}= (\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5} = (\frac{2}{13})^{-2}\cdot\big( (\frac{2}{13})^{-1}\big)^{-5} =(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5}$

In the next stage we'll recall the law of exponents for multiplication of terms with identical bases:

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

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

$(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5} =(\frac{2}{13}\big)^{-2+5}=(\frac{2}{13}\big)^{3}$

Where in the first stage we applied the above law of exponents and then simplified the expression,

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

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}= (\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5} =(\frac{2}{13})^{-2}\cdot (\frac{2}{13})^{5}=(\frac{2}{13}\big)^{3}$

Therefore the correct answer is answer A.