Simplify To Its Simplest Form: 25 36 + − 8 27 + 2 − 2 3 \sqrt{\frac{25}{36}} + \sqrt[3]{\frac{-8}{27} + 2^{-2}} 36 25 ​ ​ + 3 27 − 8 ​ + 2 − 2 ​

$$

Understanding the Problem

The given expression involves the sum of two terms, each containing a square root and a cube root. To simplify this expression, we need to evaluate each term separately and then add them together. The first term is a square root of a fraction, while the second term is a cube root of an expression involving a fraction and a negative exponent.

Simplifying the First Term

The first term is $\sqrt{\frac{25}{36}}$. To simplify this term, we can start by finding the square root of the numerator and the denominator separately.

Finding the Square Root of the Numerator

The square root of 25 is 5, since $5^2 = 25$.

Finding the Square Root of the Denominator

The square root of 36 is 6, since $6^2 = 36$.

Simplifying the First Term

Now that we have found the square roots of the numerator and the denominator, we can simplify the first term as follows:

$\sqrt{\frac{25}{36}} = \frac{\sqrt{25}}{\sqrt{36}} = \frac{5}{6}$

Simplifying the Second Term

The second term is $\sqrt[3]{\frac{-8}{27} + 2^{-2}}$. To simplify this term, we need to evaluate the expression inside the cube root.

Evaluating the Expression Inside the Cube Root

The expression inside the cube root is $\frac{-8}{27} + 2^{-2}$. To evaluate this expression, we need to find the value of $2^{-2}$.

Finding the Value of $2^{-2}$

The value of $2^{-2}$ is $\frac{1}{2^2} = \frac{1}{4}$.

Simplifying the Expression Inside the Cube Root

Now that we have found the value of $2^{-2}$, we can simplify the expression inside the cube root as follows:

$\frac{-8}{27} + 2^{-2} = \frac{-8}{27} + \frac{1}{4}$

To add these two fractions, we need to find a common denominator. The least common multiple of 27 and 4 is 108.

Finding the Common Denominator

The common denominator is 108.

Adding the Fractions

Now that we have found the common denominator, we can add the fractions as follows:

$\frac{-8}{27} + \frac{1}{4} = \frac{-8 \times 4}{27 \times 4} + \frac{1 \times 27}{4 \times 27} = \frac{-32}{108} + \frac{27}{108}$

Simplifying the Expression Inside the Cube Root

Now that we have added the fractions, we can simplify the expression inside the cube root as follows:

$\frac{-8}{27} + 2^{-2} = \frac{-32}{108} + \frac{27}{108} = \frac{-5}{108}$

Simplifying the Second Term

Now that we have simplified the expression inside the cube root, we can simplify the second term as follows:

$\sqrt[3]{\frac{-8}{27} + 2^{-2}} = \sqrt[3]{\frac{-5}{108}}$

Simplifying the Cube Root

To simplify the cube root, we can start by finding the cube root of the numerator and the denominator separately.

Finding the Cube Root of the Numerator

The cube root of -5 is -1.71 (approximately), since $(-1.71)^3 \approx -5$.

Finding the Cube Root of the Denominator

The cube root of 108 is 4.64 (approximately), since $(4.64)^3 \approx 108$.

Simplifying the Cube Root

Now that we have found the cube roots of the numerator and the denominator, we can simplify the cube root as follows:

$\sqrt[3]{\frac{-5}{108}} = \frac{\sqrt[3]{-5}}{\sqrt[3]{108}} = \frac{-1.71}{4.64}$

Adding the Two Terms

Now that we have simplified the two terms, we can add them together as follows:

$\sqrt{\frac{25}{36}} + \sqrt[3]{\frac{-8}{27} + 2^{-2}} = \frac{5}{6} + \frac{-1.71}{4.64}$

To add these two fractions, we need to find a common denominator. However, since the two fractions have different denominators, we can convert them to decimals and then add them.

Converting the Fractions to Decimals

The decimal equivalent of $\frac{5}{6}$ is 0.83.

The decimal equivalent of $\frac{-1.71}{4.64}$ is -0.37.

Adding the Decimals

Now that we have converted the fractions to decimals, we can add them as follows:

0.83 + (-0.37) = 0.46

Conclusion

In conclusion, the simplified form of the given expression is 0.46.

$$

Understanding the Problem

The given expression involves the sum of two terms, each containing a square root and a cube root. To simplify this expression, we need to evaluate each term separately and then add them together. The first term is a square root of a fraction, while the second term is a cube root of an expression involving a fraction and a negative exponent.

Q&A

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to evaluate the square root of the fraction $\frac{25}{36}$.

Q: How do we simplify the square root of a fraction?

A: To simplify the square root of a fraction, we can find the square root of the numerator and the denominator separately.

Q: What is the square root of 25?

A: The square root of 25 is 5, since $5^2 = 25$.

Q: What is the square root of 36?

A: The square root of 36 is 6, since $6^2 = 36$.

Q: How do we simplify the first term?

A: Now that we have found the square roots of the numerator and the denominator, we can simplify the first term as follows:

$\sqrt{\frac{25}{36}} = \frac{\sqrt{25}}{\sqrt{36}} = \frac{5}{6}$

Q: What is the second term in the given expression?

A: The second term is $\sqrt[3]{\frac{-8}{27} + 2^{-2}}$.

Q: How do we simplify the second term?

A: To simplify the second term, we need to evaluate the expression inside the cube root.

Q: What is the expression inside the cube root?

A: The expression inside the cube root is $\frac{-8}{27} + 2^{-2}$.

Q: How do we evaluate the expression inside the cube root?

A: To evaluate the expression inside the cube root, we need to find the value of $2^{-2}$.

Q: What is the value of $2^{-2}$?

A: The value of $2^{-2}$ is $\frac{1}{2^2} = \frac{1}{4}$.

Q: How do we simplify the expression inside the cube root?

A: Now that we have found the value of $2^{-2}$, we can simplify the expression inside the cube root as follows:

$\frac{-8}{27} + 2^{-2} = \frac{-8}{27} + \frac{1}{4}$

Q: How do we add the fractions?

A: To add the fractions, we need to find a common denominator. The least common multiple of 27 and 4 is 108.

Q: What is the common denominator?

A: The common denominator is 108.

Q: How do we add the fractions with the common denominator?

A: Now that we have found the common denominator, we can add the fractions as follows:

$\frac{-8}{27} + \frac{1}{4} = \frac{-8 \times 4}{27 \times 4} + \frac{1 \times 27}{4 \times 27} = \frac{-32}{108} + \frac{27}{108}$

Q: How do we simplify the expression inside the cube root?

A: Now that we have added the fractions, we can simplify the expression inside the cube root as follows:

$\frac{-8}{27} + 2^{-2} = \frac{-32}{108} + \frac{27}{108} = \frac{-5}{108}$

Q: How do we simplify the cube root?

A: To simplify the cube root, we can start by finding the cube root of the numerator and the denominator separately.

Q: What is the cube root of -5?

A: The cube root of -5 is -1.71 (approximately), since $(-1.71)^3 \approx -5$.

Q: What is the cube root of 108?

A: The cube root of 108 is 4.64 (approximately), since $(4.64)^3 \approx 108$.

Q: How do we simplify the cube root?

A: Now that we have found the cube roots of the numerator and the denominator, we can simplify the cube root as follows:

$\sqrt[3]{\frac{-5}{108}} = \frac{\sqrt[3]{-5}}{\sqrt[3]{108}} = \frac{-1.71}{4.64}$

Q: How do we add the two terms?

A: Now that we have simplified the two terms, we can add them together as follows:

$\sqrt{\frac{25}{36}} + \sqrt[3]{\frac{-8}{27} + 2^{-2}} = \frac{5}{6} + \frac{-1.71}{4.64}$

Q: How do we add the decimals?

A: To add the decimals, we can convert the fractions to decimals and then add them.

Q: What is the decimal equivalent of $\frac{5}{6}$?

A: The decimal equivalent of $\frac{5}{6}$ is 0.83.

Q: What is the decimal equivalent of $\frac{-1.71}{4.64}$?

A: The decimal equivalent of $\frac{-1.71}{4.64}$ is -0.37.

Q: How do we add the decimals?

A: Now that we have converted the fractions to decimals, we can add them as follows:

0.83 + (-0.37) = 0.46

Conclusion

In conclusion, the simplified form of the given expression is 0.46.