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Magazine Chapter 10: Problem 11
For each number, find all of its square roots. $$ 144 $$
Short Answer
Expert verified
The square roots of 144 are 12 and -12.
Step by step solution
01
Identify the Definition of a Square Root
A square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, for a number \( x \), a square root is \( y \) such that \( y^2 = x \).
02
Write the Equation
To find the square roots of 144, set up the equation: \( y^2 = 144 \). Here, \( y \) represents the square roots we are looking for.
03
Solve the Equation
To solve the equation \( y^2 = 144 \), take the square root of both sides. This gives two possible values: \( y = \sqrt{144} \) and \( y = -\sqrt{144} \).
04
Calculate the Positive Square Root
The positive square root of 144 is calculated as follows: \( \sqrt{144} = 12 \). So, one square root is 12.
05
Calculate the Negative Square Root
Similarly, the negative square root is calculated: \( -\sqrt{144} = -12 \). Hence, the other square root is -12.
06
List All Square Roots
The square roots of 144 are both the positive and negative values we found: 12 and -12.
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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 of a number is the non-negative value that, when multiplied by itself, gives the original number. For example, the positive square root of 144 is found by solving \( y^2 = 144 \), which gives \( y = \sqrt{144} \).
When you calculate \( \sqrt{144} \), you get 12. This is the positive square root.
The positive square root is often what is meant when we refer to 'the square root' of a number in everyday situations.
This value is always non-negative and is useful in many applications such as geometry, physics, and statistics.
When you calculate \( \sqrt{144} \), you get 12. This is the positive square root.
The positive square root is often what is meant when we refer to 'the square root' of a number in everyday situations.
This value is always non-negative and is useful in many applications such as geometry, physics, and statistics.
negative square root
In addition to the positive square root, every positive number also has a negative square root. This is the negative value that, when squared, also gives the original number. For 144, the negative square root would be calculated as \( y = -\sqrt{144} \).
- Calculated, this gives \( -\sqrt{144} = -12 \).
- So, the negative square root of 144 is -12.
solving equations
To find square roots, we often set up and solve equations. Let's take the equation \( y^2 = 144 \) as an example.
Here are the steps to solve it:
This gives you the solutions \ y= 12 \ and \ y = -12 \. These steps can be applied to any number to find its square roots.
Here are the steps to solve it:
- First, identify what the equation is asking for, which is to find y when \( y^2 = 144 \).
- Next, determine both the positive and negative values that satisfy this equation: \( \sqrt{144} \) and \(-\sqrt{144} \).
This gives you the solutions \ y= 12 \ and \ y = -12 \. These steps can be applied to any number to find its square roots.
mathematical definition
A square root of a number x is a number y such that \( y^2 = x \). This is the fundamental definition used in all calculations involving square roots.
The key points are:
The key points are:
- For any positive number, there are always two square roots: one positive and one negative.
- The positive square root is denoted as \( \sqrt{x} \), while the negative square root is represented as \( -\sqrt{x} \).
- Square roots are crucial in various fields, including geometry, where they are used to find the lengths of sides in right-angled triangles, and in statistics, where they are used to measure spread in data sets.
