cube root of 16 simplified
Understand the Problem
The question is asking for the simplified form of the cube root of 16. This involves finding a number that, when multiplied by itself three times, equals 16.
Answer
The simplified form of the cube root of 16 is $2 \sqrt[3]{2}$.
Answer for screen readers
The simplified form of the cube root of 16 is $2 \sqrt[3]{2}$.
Steps to Solve
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Express the number as a power To simplify the cube root of 16, we can express 16 as a power of 2. Since $16 = 2^4$, we can rewrite the cube root as follows: $$ \sqrt[3]{16} = \sqrt[3]{2^4} $$
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Use the properties of exponents When taking the cube root of a power, we can use the property of exponents which states that $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Therefore, we can write the expression as: $$ \sqrt[3]{2^4} = 2^{\frac{4}{3}} $$
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Rewriting the exponent The exponent $\frac{4}{3}$ can be expressed as a mixed number. This means we can write: $$ 2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}} = 2^1 \cdot 2^{\frac{1}{3}} $$
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Calculate the values Now we can substitute back to obtain the simplified expression. Calculating $2^1$ gives us 2: $$ 2^1 = 2 $$ Thus, the expression simplifies to: $$ 2 \cdot \sqrt[3]{2} $$
The simplified form of the cube root of 16 is $2 \sqrt[3]{2}$.
More Information
The cube root function finds the number that, when cubed (multiplied by itself two more times), gives the original number. In this case, expressing 16 as $2^4$ made it easier to simplify the cube root.
Tips
- Confusing the cube root with the square root. Remember, we're dealing with the cube root here.
- Not expressing 16 in terms of its prime factors, which simplifies the computation of roots.
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