What is the square root of 16384?
Understand the Problem
The question is asking for the square root of the number 16384. To find this, we would perform the mathematical operation of determining what number squared equals 16384.
Answer
$128$
Answer for screen readers
The square root of 16384 is $128$.
Steps to Solve
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Setting Up the Square Root Calculation
We need to calculate the square root of 16384. This can be expressed mathematically as:
$$ \sqrt{16384} $$ -
Finding the Square Root
We can determine the square root by finding two equal numbers which, when multiplied together, give 16384. For this, we can use calculators or prime factorization methods. -
Using Prime Factorization
Let's factor 16384 into its prime components.
- Dividing by 2 repeatedly:
$$ 16384 \div 2 = 8192 $$
$$ 8192 \div 2 = 4096 $$
$$ 4096 \div 2 = 2048 $$
$$ 2048 \div 2 = 1024 $$
$$ 1024 \div 2 = 512 $$
$$ 512 \div 2 = 256 $$
$$ 256 \div 2 = 128 $$
$$ 128 \div 2 = 64 $$
$$ 64 \div 2 = 32 $$
$$ 32 \div 2 = 16 $$
$$ 16 \div 2 = 8 $$
$$ 8 \div 2 = 4 $$
$$ 4 \div 2 = 2 $$
$$ 2 \div 2 = 1 $$
This gives us:
$$ 16384 = 2^{14} $$
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Calculating the Square Root from Prime Factorization
Now, we use the property of square roots:
$$ \sqrt{a^{2}} = a $$
Thus, we can find:
$$ \sqrt{16384} = \sqrt{2^{14}} = 2^{14/2} = 2^{7} = 128 $$
The square root of 16384 is $128$.
More Information
Finding square roots is an essential mathematical skill used in various applications, from geometry to algebra. Knowing that $16384 = 2^{14}$ helps speed up the calculation and provides insights into the properties of numbers.
Tips
- Incorrect Factorization: Failing to factor correctly or counting errors when dividing can lead to the wrong square root.
- Ignoring Prime Factorization: Skipping prime factorization may make it harder to recognize perfect squares.
