Powers: Power of a fraction

To solve this problem, we will follow these steps:

• Step 1: Square the numerator $3$.
• Step 2: Square the denominator $2$.
• Step 3: Simplify the resulting fraction if needed.

Let's solve the problem:

Step 1: The numerator is $3$. Squaring $3$ gives us:

$3^2 = 9$

Step 2: The denominator is $2$. Squaring $2$ gives us:

$2^2 = 4$

Step 3: We now write the fraction as:

$\left(\frac{3}{2}\right)^2 = \frac{9}{4}$

To express $\frac{9}{4}$ as a mixed number, we divide $9$ by $4$:

$9$ divided by $4$ is $2$ with a remainder of $1$. Thus, $\frac{9}{4}$ can be expressed as:

$2\frac{1}{4}$

Therefore, the solution to the problem is:

$2\frac{1}{4}$

The correct choice according to given possible answers is choice 4.

To solve for $\left( \frac{1}{3} \right)^2$, we follow these steps:

• Step 1: Identify the given fraction, which is $\frac{1}{3}$.
• Step 2: Apply the formula for squaring a fraction: $\left( \frac{a}{b} \right)^2 = \frac{a^2}{b^2}$.
• Step 3: Square the numerator: $1^2 = 1$.
• Step 4: Square the denominator: $3^2 = 9$.
• Step 5: Form the new fraction using the squared values: $\frac{1^2}{3^2} = \frac{1}{9}$.

Therefore, the solution to $\left( \frac{1}{3} \right)^2$ is $\frac{1}{9}$.

To solve the expression $\left(\frac{1}{3}\right)^3$, we will apply the power of a fraction rule.

Step 1: Begin with the expression $\left(\frac{1}{3}\right)^3$.
This means we need to calculate $\frac{1^3}{3^3}$.

Step 2: Evaluate $1^3$ and $3^3$:
- $1^3 = 1 \times 1 \times 1 = 1$
- $3^3 = 3 \times 3 \times 3 = 27$

Step 3: Construct the fraction with these powers:
$\frac{1^3}{3^3} = \frac{1}{27}$.

Therefore, the value of $\left(\frac{1}{3}\right)^3$ is $\frac{1}{27}$.

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

• Step 1: Identify the given fraction, which is $\frac{2}{3}$.
• Step 2: Apply the formula for exponents applied to fractions, $\displaystyle(\frac{a}{b})^n = \frac{a^n}{b^n}$.
• Step 3: Calculate the cube of the numerator and the cube of the denominator separately.
• Step 4: Write the results as a single fraction.

Now, let's work through each step:

Step 1: The problem provides the fraction $\frac{2}{3}$.

Step 2: Use the formula $(\frac{a}{b})^n = \frac{a^n}{b^n}$ for $a = 2$, $b = 3$, and $n = 3$.

Step 3: We need to calculate $2^3$ and $3^3$:
- Calculate $2^3 = 2 \times 2 \times 2 = 8$.
- Calculate $3^3 = 3 \times 3 \times 3 = 27$.

Step 4: Writing these as a fraction gives $\displaystyle \frac{2^3}{3^3} = \frac{8}{27}$.

Therefore, the solution to the problem is $\frac{8}{27}$.