Question

Which of the options below is the fully simplified form of √128 ?
√128
2√32
4√8
8√2

Answer 1

explain your thinking 9th grade response

Answer 2

To simplify \(\sqrt{128}\), we can start by breaking 128 down into its prime factors.

128 can be expressed as:
\[
128 = 64 \times 2 = 8^2 \times 2 = (4^2 \times 2) = (2^6)
\]

Now, we can simplify \(\sqrt{128}\):
\[
\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}
\]

So the fully simplified form of \(\sqrt{128}\) is:
\[
\boxed{8\sqrt{2}}
\]

To verify:

1. \(\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \cdot \sqrt{2} = 8\sqrt{2}\)
2.

Thus, the correct answer from the options given is \(8\sqrt{2}\).

Answer 3

Sure! Let's break down how we simplify \(\sqrt{128}\) step by step in a way that's easy to understand.

### Step 1: Understand Square Roots
A square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

### Step 2: Factor the Number Under the Square Root
To simplify \(\sqrt{128}\), we want to break down 128 into simpler factors. We can start by looking for perfect squares (numbers that are squares of whole numbers).

### Step 3: Finding Perfect Squares
Let's find the factors of 128:
- We can divide it by 2 repeatedly:
- \(128 \div 2 = 64\)
- \(64 \div 2 = 32\)
- \(32 \div 2 = 16\)
- \(16 \div 2 = 8\)
- \(8 \div 2 = 4\)
- \(4 \div 2 = 2\)
- \(2 \div 2 = 1\)

So, the complete factorization of 128 is:
\[
128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7
\]
Another way to think about this is that 128 can also be expressed as \(64 \times 2\), and 64 is a perfect square because \(64 = 8 \times 8\).

### Step 4: Simplifying the Square Root
Now we can simplify \(\sqrt{128}\) using our factorization. Since we can write 128 as \(64 \times 2\), we can apply the square root to both factors:
\[
\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2}
\]

### Step 5: Calculate the Square Root of the Perfect Square
We know that \(\sqrt{64} = 8\) because \(8 \times 8 = 64\). So we get:
\[
\sqrt{128} = 8 \times \sqrt{2}
\]

### Final Result
Therefore, the simplified form of \(\sqrt{128}\) is:
\[
8\sqrt{2}
\]

### Conclusion
So, from the options you provided, the fully simplified form of \(\sqrt{128}\) is \(8\sqrt{2}\). This means when you see \(\sqrt{128}\), you can rewrite it as \(8\) times the square root of \(2\), which is much simpler!