Problem 90 Determine the exact answer in si... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 1: Problem 90

Determine the exact answer in simplified form. $$ -\operatorname{sqrt}(86) $$

Short Answer

Expert verified

The simplified form is \$-\sqrt{86}\$.

Step by step solution

Understanding the Expression

The expression given is $-\sqrt{86}$. This represents the negative square root of the number 86. We need to find the simplified form of this expression.

Simplifying the Expression

Since 86 is not a perfect square and doesn't have a square factor (other than 1), $-\sqrt{86}$ is already in its simplest form.

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Key Concepts

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

Negative Square Root

A negative square root might initially seem a bit confusing, especially since when we think of square roots, we typically picture positive values. But what does it really mean? In essence, a negative square root is simply the negative of a square root of a number. If you see an expression like $-\sqrt{86}$, it means you take the square root of 86 and then multiply it by -1. This is just a way of signifying that the result is less than zero.

Some key points to remember about negative square roots:

They are not about the square root of a negative number (e.g., there is no real number that is the square root of -25; this would involve an imaginary number).
The negative sign in front simply changes the sign of the result of the square root.
You can think of it as shifting any positive result of a square root to the negative zone on the number line.

Remember, a negative square root does not change the property of the number we are taking the square root of; it only affects the sign of the overall result.

Perfect Square

Understanding the concept of perfect squares helps immensely when simplifying square roots. A perfect square is a number that is the product of an integer with itself. For example, 1, 4, 9, 16, and 25 are all perfect squares because they arise from squaring whole numbers (1x1, 2x2, 3x3, and so on).

Here鈥檚 why perfect squares are important:

They help in easily simplifying square roots because the square root of a perfect square is irreducible to another basic number; it is a whole number.
They serve as a reference point validating whether a number can be broken down into smaller factors that are perfect squares.
While working with expressions like $-\sqrt{86}$, noticing that 86 is not a perfect square tells us there is no integer that squared will yield 86.

In the case of numbers that aren't perfect squares, like 86 in our initial expression, we realize that the expression can't be further simplified into lower terms of integer squares.

Simplified Form

When we talk about a number being in its simplified form, especially with square roots, we mean that there are no perfect square factors other than 1. The goal of simplifying square roots is to make an expression as simple as possible by eliminating any factors that can be exactly squared.

Considerations when simplifying square roots:

Check if the number is a perfect square or contains square factors. If it is or does, break it down into such factors and simplify.
For a number like 86, none of its factors are perfect squares besides 1, so its square root is already simplified.
For numbers with large factors, simplification might involve recognizing \/ or dividing by smaller perfect squares (4, 9, 16, etc.) where applicable.

In cases like $-\sqrt{86}$, where the number is already in simple non-factorizable form, it remains as is. This simplicity aids understanding and continues mathematically accurate calculations in more complex algebraic forms.

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91影视

Determine the exact answer in simplified form. $$ -\operatorname{sqrt}(86) $$

Short Answer

Step by step solution

Understanding the Expression

Simplifying the Expression

Key Concepts

Negative Square Root

Perfect Square

Simplified Form

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