Question
Find the square root of 128.(1 point)
Responses
42–√
4 start root 2 end root
24–√
2 start root 4 end root
82–√
8 start root 2 end root
64 −−−√ x 2–√
start root 64 end root x start root 2 end root
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The correct response is 8√2.
To find the square root of 128, you can use the method of prime factorization and the property of square roots.
First, we need to break down the number 128 into its prime factors. The prime factorization of 128 can be found by dividing it by the smallest prime number, which is 2. Repeatedly dividing 128 by 2 until we can't split it any further, we get:
128 ÷ 2 = 64
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
So the prime factorization of 128 is 2 × 2 × 2 × 2 × 2 × 2 = 2^7.
Now, let's pair up the prime factors in groups of two, since a square root should have pairs of the same numbers. We have (2 × 2) × (2 × 2) × (2 × 2) = 2^3 × 2^3 × 2^1.
Using the property of square roots, we can take one factor of 2 from each pair and get a simplified expression under the square root:
√(2^3 × 2^3 × 2^1) = √(2^3) × √(2^3) × √(2^1) = 2^3 × 2^3 × 2^0.5
Simplifying further:
= 2^(3 + 3 + 0.5) = 2^6.5
Therefore, the square root of 128 is 2^6.5.
To find the square root of 128, we can break it down into smaller factors.
Step 1: Since 128 can be simplified, let's start by factoring it.
128 = 64 * 2
Step 2: Now, let's find the square root of the factors separately.
The square root of 64 is 8. So, √64 = 8
The square root of 2 cannot be simplified further, so we leave it as is.
Combining the results from the two factors, we have:
√128 = 8√2
Therefore, the square root of 128 is 8√2.