The exact numbers that, when multiplied by themselves, equal 343 are the positive and negative square roots of 343, precisely expressed as √343 and -√343.
Understanding the Concept
When a question asks "what times itself equals a number?", it is fundamentally asking for the square root of that number. A square root of a number is a value that, when multiplied by itself (or "squared"), gives the original number. For example, since 5 × 5 = 25, the positive square root of 25 is 5.
The square root of 343 is symbolically represented as √343. Every positive number has two real square roots: one positive and one negative.
Exact vs. Approximate Values
Since 343 is not a perfect square (a number whose square root is a whole number, like 25 or 400), its square root is an irrational number. This means its decimal representation goes on infinitely without repeating, making it impossible to write down precisely in decimal form. Therefore, the most accurate way to express the answer is using the radical symbol.
However, for practical purposes, an approximate decimal value can be used. A number that, when multiplied by itself, gives the original number 343 is its square root. When multiplied by itself, the number 18.520 approximately equals 343. This means that both positive and negative values need to be considered:
- Positive Root: √343 ≈ 18.520
- Negative Root: -√343 ≈ -18.520
So, if you multiply 18.520 by 18.520, you will get a number very close to 343 (approximately 342.9904). Similarly, multiplying -18.520 by -18.520 also yields approximately 342.9904.
Summary of the Answers
To clearly differentiate between the precise mathematical answer and its common approximation, consider the following:
| Aspect | Value | Description |
|---|---|---|
| Exact Values | √343 and -√343 | The mathematically precise answers, representing the positive and negative square roots in radical form. |
| Approximate Values | ≈ 18.520 and ≈ -18.520 | Decimal approximations for practical calculations, as 343 is not a perfect square and its decimal representation is non-terminating and non-repeating. |
Mathematical Representation
In algebra, if we let 'x' be the number that times itself equals 343, the equation is:
x² = 343
To solve for 'x', we take the square root of both sides:
x = ±√343
This notation acknowledges both the positive and negative roots. For more information on square roots and irrational numbers, you can refer to mathematical resources on square roots or irrational numbers.
