What is the other square root of 119 + 120
The Correct Answer and Explanation is:
The expression is:
119 + 120 = 239
Now, we are asked to find the other square root of 239.
Step 1: Understanding Square Roots
The square root of a number xxx is any number yyy such that y2=xy^2 = xy2=x. For any positive real number like 239, there are always two square roots:
- A positive square root: 239\sqrt{239}239
- A negative square root: −239-\sqrt{239}−239
So, the square roots of 239 are:239and−239\sqrt{239} \quad \text{and} \quad -\sqrt{239}239and−239
The phrase “the other square root” refers to the square root that is not commonly given. In most mathematical contexts, 239\sqrt{239}239 refers only to the positive root. So, the other one is the negative root:−239\boxed{-\sqrt{239}}−239
This is the correct answer.
Explanation
Every positive real number has exactly two real square roots. One is positive and the other is negative. For example, the square roots of 9 are 3 and -3 because both 32=93^2 = 932=9 and (−3)2=9(-3)^2 = 9(−3)2=9. This is a fundamental concept in algebra.
When we take the square root of a number, such as 239\sqrt{239}239, we typically refer only to the principal square root, which is the positive value. However, there is always a second real number that also satisfies the equation x2=239x^2 = 239×2=239, and that is the negative root, −239-\sqrt{239}−239.
In this particular problem, the sum of 119 and 120 gives 239. The question then asks for the other square root of that result. Since 239\sqrt{239}239 is the principal root, the “other” root must be −239-\sqrt{239}−239. This approach relies on a clear understanding of how square roots work and how equations like x2=239x^2 = 239×2=239 have two solutions.
It is important to note that 239 is not a perfect square. That means its square roots cannot be expressed as exact integers. Approximating with a calculator gives:239≈15.4596\sqrt{239} \approx 15.4596239≈15.4596
Therefore:−239≈−15.4596-\sqrt{239} \approx -15.4596−239≈−15.4596
So, the final answer is −239\boxed{-\sqrt{239}}−239, or approximately −15.4596\boxed{-15.4596}−15.4596.
