https://file.aiquickdraw.com/mathbot/file/1726591581120exh634o1.jpg

Let's simplify the given expression step by step. The expression is:

\[
\frac{\sqrt{25}}{\sqrt[3]{64}} + \left(\frac{256}{625}\right)^{-1/4} + \frac{1}{\left(\frac{64}{25}\right)^{2/3}}
\]

### Step 1: Simplifying the first term
\[
\frac{\sqrt{25}}{\sqrt[3]{64}} = \frac{5}{4}
\]
- \(\sqrt{25} = 5\) since 5 is the square root of 25.
- \(\sqrt[3]{64} = 4\) since \(4^3 = 64\).

### Step 2: Simplifying the second term
\[
\left(\frac{256}{625}\right)^{-1/4}
\]
- First, calculate the fourth root: \(\sqrt[4]{256} = 4\) (since \(4^4 = 256\)) and \(\sqrt[4]{625} = 5\) (since \(5^4 = 625\)).
- The expression becomes \(\left(\frac{4}{5}\right)^{-1}\), which is the reciprocal: \(\frac{5}{4}\).

### Step 3: Simplifying the third term
\[
\frac{1}{\left(\frac{64}{25}\right)^{2/3}}
\]
- First, take the cube root: \(\sqrt[3]{64} = 4\) and \(\sqrt[3]{25} = 5\).
- The expression becomes \(\frac{4}{5}\).
- Now square this: \(\left(\frac{4}{5}\right)^2 = \frac{16}{25}\).
- The reciprocal is: \(\frac{1}{\frac{16}{25}} = \frac{25}{16}\).

### Step 4: Adding all terms
Now sum the results:
\[
\frac{5}{4} + \frac{5}{4} + \frac{25}{16}
\]
- Convert all fractions to have a common denominator of 16:
\[
\frac{5}{4} = \frac{20}{16}
\]
- Adding the fractions:
\[
\frac{20}{16} + \frac{20}{16} + \frac{25}{16} = \frac{65}{16}
\]

Thus, the simplified expression is:

\[
\boxed{\frac{65}{16}}
\]

Would you like more details or explanations on any step? Here are some related questions to explore further:

1. How do you find the reciprocal of a fraction raised to a negative exponent?
2. What is the cube root of a number, and how do you calculate it?
3. How can you add fractions with different denominators efficiently?
4. How do you evaluate fractional exponents like \( x^{2/3} \)?
5. Why does raising a fraction to a negative exponent involve taking the reciprocal?

**Tip**: When simplifying roots and exponents, it's often helpful to rewrite the terms in fractional form for easier manipulation.

Simplify: (√25 / ∛64) + (256/625)^(-1/4) + 1 / (64/25)^(2/3)

Step-by-step solution to simplify the expression √25 / ∛64 + (256/625)^(-1/4) + 1/(64/25)^(2/3). This problem covers important algebraic concepts such as roots, exponents, and simplifying fractions. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation