Solved on Dec 04, 2023

Solve for the missing values in the given equations: $\frac{-5}{2^{2}}=\square$ and $\left(-\frac{2}{3}\right)^{3}=\square$ .

STEP 1

Assumptions
1. We need to evaluate the expression $\frac{-5}{2^{2}}$ .
2. We need to evaluate the expression $\left(-\frac{2}{3}\right)^{3}$ .

STEP 2

First, we will solve the expression $\frac{-5}{2^{2}}$ . We need to calculate the square of 2, which is the denominator.
$2^{2} = 2 \times 2$

STEP 3

Calculate the square of 2.
$2^{2} = 4$

STEP 4

Now, divide the numerator by the square of 2.
$\frac{-5}{2^{2}} = \frac{-5}{4}$

STEP 5

The expression simplifies to the fraction $\frac{-5}{4}$ . This is the final answer for the first expression.
$\frac{-5}{2^{2}} = \frac{-5}{4}$

STEP 6

Next, we will solve the expression $\left(-\frac{2}{3}\right)^{3}$ . We need to raise the fraction $-\frac{2}{3}$ to the power of 3.
$\left(-\frac{2}{3}\right)^{3} = -\frac{2}{3} \times -\frac{2}{3} \times -\frac{2}{3}$

STEP 7

First, calculate the square of $-\frac{2}{3}$ .
$\left(-\frac{2}{3}\right)^{2} = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)$

STEP 8

Calculate the square of $-\frac{2}{3}$ .
$\left(-\frac{2}{3}\right)^{2} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}$

STEP 9

Now, multiply the result by $-\frac{2}{3}$ to get the cube.
$\left(-\frac{2}{3}\right)^{3} = \frac{4}{9} \times -\frac{2}{3}$

STEP 10

Multiply the numerators and the denominators separately.
$\left(-\frac{2}{3}\right)^{3} = \frac{4 \times -2}{9 \times 3}$

STEP 11

Calculate the multiplication.
$\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$

STEP 12

The expression simplifies to the fraction $\frac{-8}{27}$ . This is the final answer for the second expression.
$\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$
The solutions are: $\frac{-5}{2^{2}} = \frac{-5}{4}$ $\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$

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Solve for the missing values in the given equations: $\frac{-5}{2^{2}}=\square$ and $\left(-\frac{2}{3}\right)^{3}=\square$ .

STEP 1

Assumptions
1. We need to evaluate the expression $\frac{-5}{2^{2}}$ .
2. We need to evaluate the expression $\left(-\frac{2}{3}\right)^{3}$ .

STEP 2

First, we will solve the expression $\frac{-5}{2^{2}}$ . We need to calculate the square of 2, which is the denominator.
$2^{2} = 2 \times 2$

STEP 3

Calculate the square of 2.
$2^{2} = 4$

STEP 4

Now, divide the numerator by the square of 2.
$\frac{-5}{2^{2}} = \frac{-5}{4}$

STEP 5

The expression simplifies to the fraction $\frac{-5}{4}$ . This is the final answer for the first expression.
$\frac{-5}{2^{2}} = \frac{-5}{4}$

STEP 6

Next, we will solve the expression $\left(-\frac{2}{3}\right)^{3}$ . We need to raise the fraction $-\frac{2}{3}$ to the power of 3.
$\left(-\frac{2}{3}\right)^{3} = -\frac{2}{3} \times -\frac{2}{3} \times -\frac{2}{3}$

STEP 7

First, calculate the square of $-\frac{2}{3}$ .
$\left(-\frac{2}{3}\right)^{2} = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)$

STEP 8

Calculate the square of $-\frac{2}{3}$ .
$\left(-\frac{2}{3}\right)^{2} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}$

STEP 9

Now, multiply the result by $-\frac{2}{3}$ to get the cube.
$\left(-\frac{2}{3}\right)^{3} = \frac{4}{9} \times -\frac{2}{3}$

STEP 10

Multiply the numerators and the denominators separately.
$\left(-\frac{2}{3}\right)^{3} = \frac{4 \times -2}{9 \times 3}$

STEP 11

Calculate the multiplication.
$\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$

STEP 12

The expression simplifies to the fraction $\frac{-8}{27}$ . This is the final answer for the second expression.
$\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$
The solutions are: $\frac{-5}{2^{2}} = \frac{-5}{4}$ $\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$

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Solve for the missing values in the given equations: −522=□\frac{-5}{2^{2}}=\square22−5​=□ and (−23)3=□\left(-\frac{2}{3}\right)^{3}=\square(−32​)3=□.

STEP 1

Assumptions1. We need to evaluate the expression −522\frac{-5}{2^{2}}22−5​.2. We need to evaluate the expression (−23)3\left(-\frac{2}{3}\right)^{3}(−32​)3.

STEP 2

First, we will solve the expression −522\frac{-5}{2^{2}}22−5​. We need to calculate the square of 2, which is the denominator.22=2×22^{2} = 2 \times 222=2×2

STEP 3

Calculate the square of 2.22=42^{2} = 422=4

STEP 4

Now, divide the numerator by the square of 2.−522=−54\frac{-5}{2^{2}} = \frac{-5}{4}22−5​=4−5​

STEP 5

The expression simplifies to the fraction −54\frac{-5}{4}4−5​. This is the final answer for the first expression.−522=−54\frac{-5}{2^{2}} = \frac{-5}{4}22−5​=4−5​

STEP 6

STEP 7

First, calculate the square of −23-\frac{2}{3}−32​.(−23)2=(−23)×(−23)\left(-\frac{2}{3}\right)^{2} = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)(−32​)2=(−32​)×(−32​)

STEP 8

Calculate the square of −23-\frac{2}{3}−32​.(−23)2=2×23×3=49\left(-\frac{2}{3}\right)^{2} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}(−32​)2=3×32×2​=94​

STEP 9

Now, multiply the result by −23-\frac{2}{3}−32​ to get the cube.(−23)3=49×−23\left(-\frac{2}{3}\right)^{3} = \frac{4}{9} \times -\frac{2}{3}(−32​)3=94​×−32​

STEP 10

Multiply the numerators and the denominators separately.(−23)3=4×−29×3\left(-\frac{2}{3}\right)^{3} = \frac{4 \times -2}{9 \times 3}(−32​)3=9×34×−2​

STEP 11

Calculate the multiplication.(−23)3=−827\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}(−32​)3=27−8​

STEP 12

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Solve for the missing values in the given equations: $\frac{-5}{2^{2}}=\square$ and $\left(-\frac{2}{3}\right)^{3}=\square$ .

Assumptions
1. We need to evaluate the expression $\frac{-5}{2^{2}}$ .
2. We need to evaluate the expression $\left(-\frac{2}{3}\right)^{3}$ .

First, we will solve the expression $\frac{-5}{2^{2}}$ . We need to calculate the square of 2, which is the denominator.
$2^{2} = 2 \times 2$

Calculate the square of 2.
$2^{2} = 4$

Now, divide the numerator by the square of 2.
$\frac{-5}{2^{2}} = \frac{-5}{4}$

The expression simplifies to the fraction $\frac{-5}{4}$ . This is the final answer for the first expression.
$\frac{-5}{2^{2}} = \frac{-5}{4}$

First, calculate the square of $-\frac{2}{3}$ .
$\left(-\frac{2}{3}\right)^{2} = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)$

Calculate the square of $-\frac{2}{3}$ .
$\left(-\frac{2}{3}\right)^{2} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}$

Now, multiply the result by $-\frac{2}{3}$ to get the cube.
$\left(-\frac{2}{3}\right)^{3} = \frac{4}{9} \times -\frac{2}{3}$

Multiply the numerators and the denominators separately.
$\left(-\frac{2}{3}\right)^{3} = \frac{4 \times -2}{9 \times 3}$

Calculate the multiplication.
$\left(-\frac{2}{3}\right)^{3} = \frac{-8}{27}$