What is the value of `(""^(4) sqrt(16)+""^(4) sqrt(625))/(""^(4) sqrt(256))`?

Options

A. `1.75`
B. `1.25`
C. `1.6`
D. `1.5`

Answer

To solve the expression \((\sqrt[4]{16} + \sqrt[4]{625}) / \sqrt[4]{256}\), let's break it down step by step.

### Step 1: Calculate \(\sqrt[4]{16}\)
We know that \(16\) can be expressed as \(2^4\).
\[ \sqrt[4]{16} = (2^4)^{1/4} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \]

**Hint:** Remember that \(\sqrt[n]{a^m} = a^{m/n}\).

### Step 2: Calculate \(\sqrt[4]{625}\)
Next, we express \(625\) as \(5^4\).
\[ \sqrt[4]{625} = (5^4)^{1/4} = 5^{4 \cdot \frac{1}{4}} = 5^1 = 5 \]

**Hint:** Identify perfect powers to simplify the roots easily.

### Step 3: Calculate \(\sqrt[4]{256}\)
Now, we express \(256\) as \(2^8\).
\[ \sqrt[4]{256} = (2^8)^{1/4} = 2^{8 \cdot \frac{1}{4}} = 2^2 = 4 \]

**Hint:** Again, look for perfect powers to simplify calculations.

### Step 4: Substitute the values into the expression
Now we can substitute the values we found back into the expression:
\[ \frac{\sqrt[4]{16} + \sqrt[4]{625}}{\sqrt[4]{256}} = \frac{2 + 5}{4} \]

### Step 5: Simplify the expression
Now we simplify the numerator:
\[ 2 + 5 = 7 \]
Thus, we have:
\[ \frac{7}{4} \]

### Step 6: Convert to decimal (if needed)
To convert \(\frac{7}{4}\) to decimal:
\[ \frac{7}{4} = 1.75 \]

### Final Answer
The value of \(\frac{\sqrt[4]{16} + \sqrt[4]{625}}{\sqrt[4]{256}}\) is \(1.75\).

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