What is the 6th root of 729?
Understand the Problem
The question is asking for the 6th root of the number 729, which involves finding a number that, when raised to the power of 6, equals 729.
Answer
$3$
Answer for screen readers
The 6th root of 729 is 3.
Steps to Solve
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Identify the exponent and the base We need to find a number that raised to the power of 6 equals 729. This means we are looking for the number $x$ such that $x^6 = 729$.
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Rewriting the equation We can rewrite the equation to isolate $x$ by taking the 6th root of both sides. This gives us:
$$ x = 729^{1/6} $$
- Calculating the value of 729 Next, let's break down the number 729. We can factor it into prime factors:
$$ 729 = 3^6 $$
- Taking the 6th root Now that we know that $729 = 3^6$, we can take the 6th root:
$$ x = (3^6)^{1/6} $$
- Simplifying the expression Using the property of exponents $(a^m)^{n} = a^{m \cdot n}$, we can simplify:
$$ x = 3^{6 \cdot \frac{1}{6}} = 3^1 = 3 $$
The 6th root of 729 is 3.
More Information
The number 729 is a perfect sixth power of 3, which can be useful in various mathematical contexts. Perfect powers often appear in number theory and problems involving roots.
Tips
- Not simplifying the base: Sometimes, individuals may try to directly compute the 6th root without recognizing that 729 can be expressed as a power of a smaller base.
- Miscalculating the root: Always ensure to check calculations step by step, especially when dealing with roots and powers.
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