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Chapter 7: Problem 14

Find all of its square roots. $$ 2500 $$

Short Answer

Expert verified
The square roots of 2500 are 50 and -50.

Step by step solution

01

- Understand Square Roots

The square root of a number is a value that, when multiplied by itself, gives the number. Mathematically, for a number \( x \), its square roots are the values \( y \) such that \( y^2 = x \).
02

- Calculate the Positive Square Root

To find the positive square root of 2500, look for a number \( y \) such that \( y^2 = 2500 \). Since 50 multiplied by 50 equals 2500, the positive square root of 2500 is 50.
03

- Calculate the Negative Square Root

In addition to the positive square root, every positive number has a negative square root. For 2500, the negative square root will be -50, since \( (-50) \times (-50) = 2500 \).
04

- List Both Square Roots

Combining both results, the square roots of 2500 are 50 and -50.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

positive square root
The positive square root is one of the two types of square roots a number can have.
  • It is the value that, when multiplied by itself, equals the original number.
  • For instance, if we take the number 2500, the positive square root is a number, let's call it y, where \( y^2 = 2500 \).
  • After calculating, we find that \( y = 50 \), because \( 50 \times 50 = 2500 \). This gives us the positive square root of 2500.
negative square root
A number also has a negative square root.
  • This is the value that, when multiplied by itself, also equals the original number, just like the positive square root, but with a different sign.
  • Using our previous example, the negative square root of 2500 is the number, let's call it -y, where \( (-y)^2 = 2500 \).
  • Since we know the positive root is 50, the negative root will be -50. Hence, \( (-50) \times (-50) = 2500 \).
  • Combining both results, 2500 has two square roots: 50 and -50.
multiplication
Understanding multiplication is crucial when dealing with square roots.
  • Multiplication is a basic arithmetic operation where a number is added to itself a certain number of times.
  • This is relevant for square roots because they represent repeated multiplication.
  • For example, if we take the positive square root of 2500, which is 50, and multiply 50 with itself, we get 2500: \( 50 \times 50 = 2500 \).
  • Similarly, for the negative square root, \( (-50) \times (-50) = 2500 \).
  • Remember that multiplying two negative numbers always results in a positive number. This explains why \( (-50) \times (-50) \) also equals 2500.

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