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Chapter 5: Problem 53

In Exercises 53-70, compute the exact value of the given expression. \(6-\sqrt{576}\)

Short Answer

Expert verified
The exact value of the expression is -18.

Step by step solution

01

Identify the square root

The expression given is \(6 - \sqrt{576}\). The first step is to find the exact value of the square root part, \(\sqrt{576}\).
02

Compute the square root

We need to calculate \(\sqrt{576}\). We know that \(24 \times 24 = 576\), hence \(\sqrt{576} = 24\).
03

Substitute back into the expression

Now replace \(\sqrt{576}\) with \(24\) in the original expression. So, we have \(6 - \sqrt{576} = 6 - 24\).
04

Calculate the final value

Subtract \(24\) from \(6\) to find the final result: \(6 - 24 = -18\).

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Key Concepts

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

Understanding Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. In simpler terms, if you take the square root of 576, you are looking for a number which, when multiplied by itself, equals 576.
In mathematical notation, this is expressed with the square root symbol \( \sqrt{} \). For example, \( \sqrt{576} = 24 \). When you see \( \sqrt{576} \), you're asked to find what squared (itself multiplied by itself) equals 576.
Knowing that 24 times 24 is 576 helps us identify \( \sqrt{576} = 24 \). Square roots often play a crucial role in simplifying expressions and solving equations, especially in geometry and algebra.
  • The square root operation is the opposite of squaring a number.
  • In mathematics, symbolically, this is often represented as \( a^2 = b \) and the reverse \( \sqrt{b} = a \).
Expression Evaluation
Expression evaluation is the process of finding the value of a mathematical expression. It involves simplifying components of the expression step by step until you reach a final result.
In the expression \( 6 - \sqrt{576} \), the first task is to evaluate the square root \( \sqrt{576} \) which we have already identified as 24. Once this is done, replace \( \sqrt{576} \) with 24 in the expression.
Now, the expression becomes \( 6 - 24 \). The culmination of expression evaluation is simplifying until it's possible to write the expression as a single numerical value.
  • Always start evaluating from the innermost expression or operation.
  • Identify any operations that can be directly computed, like square roots, to simplify the process.
Basic Mathematical Operations
Mathematical operations are the building blocks for solving expressions and equations. They include addition, subtraction, multiplication, and division. Each operation follows specific rules, namely the order of operations, which guides the correct sequence to perform calculations.
In the expression \(6 - 24\), subtraction is being performed. Here, the number 24 is being subtracted from the number 6. This results in -18, as performing the operation \(6 - 24\) requires starting from 6 and moving 24 units leftward on a number line.
  • Subtraction is finding the difference between numbers. In the equation \( a - b \), it means removing b from a.
  • Always work from left to right while performing operations.
Understanding these operations will help in accurately evaluating more complex mathematical expressions.

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