What is the cube root of 32768?
Understand the Problem
The question is asking for the cube root of the number 32768. This involves finding a number that, when multiplied by itself three times (raised to the power of 3), equals 32768.
Answer
The cube root of 32768 is \( 32 \).
Answer for screen readers
The cube root of 32768 is ( 32 ).
Steps to Solve
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Identify the problem We need to find the cube root of 32768. This can be expressed mathematically as finding ( x ) such that ( x^3 = 32768 ).
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Express using roots The cube root can be represented using fractional exponents. The equation can be written as: $$ x = 32768^{\frac{1}{3}} $$
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Estimate cube root Before calculating, we can estimate by looking for perfect cubes around 32768. The cube of 30 is ( 30^3 = 27000 ) and the cube of 40 is ( 40^3 = 64000 ). Hence, the cube root should be between 30 and 40.
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Calculate the exact cube root Now we can calculate: $$ x = 32768^{\frac{1}{3}} $$ Using a calculator, we find: $$ x = 32 $$
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Verify the solution To ensure our solution is correct, we can verify by cubing 32: $$ 32^3 = 32 \times 32 \times 32 = 32768 $$
The cube root of 32768 is ( 32 ).
More Information
The number 32 is significant as it is a power of 2, specifically ( 2^5 ). This means that ( 32768 ) can also be expressed as ( (2^5)^3 = 2^{15} ), showing the relationship between exponents and roots.
Tips
- Confusing cube roots with square roots. Be sure to understand that cube roots involve multiplying a number by itself three times, not two.
- Not verifying the result by cubing the answer. Always check your calculations to confirm the answer is correct.
