Problem 7
Question
For the following exercises, simplify each expression. \(\sqrt{4(9+16)}\)
Step-by-Step Solution
Verified Answer
The simplified expression is 10.
1Step 1: Solve Inside the Parentheses
Start by solving the expression inside the parentheses. We have \(9 + 16\). Add these two numbers together: \(9 + 16 = 25\).
2Step 2: Multiply by the Coefficient
Now take the result from Step 1 and multiply it by the coefficient that is outside the parentheses. We have \(4 imes 25\). Calculate this to get \(100\).
3Step 3: Simplify the Square Root
Now, simplify the square root of the result from Step 2. We need to find \(\sqrt{100}\). Since 100 is a perfect square, the square root of 100 is \(10\).
Key Concepts
Square RootPerfect SquaresOrder of Operations
Square Root
The square root is a fundamental concept in mathematics. When you see the square root symbol, \sqrt{}, it means you are looking for a number which, when multiplied by itself, gives back the "inside" number. This "inside" number is known as the radicand.
Take the example of our exercise, where we're asked to simplify \sqrt{100}. This expression is asking us: "What number, when squared, results in 100?"
The answer is 10, because 10 \(\times\) 10 = 100. Therefore, the square root of 100 is 10. Calculating square roots can be straightforward for numbers that are perfect squares. However, for non-perfect squares, it often requires approximation or a calculator. Underlying this is a deep understanding of how multiplication and division bind the concept of squares and their roots.
Take the example of our exercise, where we're asked to simplify \sqrt{100}. This expression is asking us: "What number, when squared, results in 100?"
The answer is 10, because 10 \(\times\) 10 = 100. Therefore, the square root of 100 is 10. Calculating square roots can be straightforward for numbers that are perfect squares. However, for non-perfect squares, it often requires approximation or a calculator. Underlying this is a deep understanding of how multiplication and division bind the concept of squares and their roots.
Perfect Squares
Understanding perfect squares can make simplifying expressions easier. A perfect square results when a whole number is multiplied by itself. These numbers include 1, 4, 9, 16, 25, and so on.
When we talk about simplifying expressions, noticing these perfect squares can save time and effort. In our example, after doing some multiplication and addition, we ended up with \(100\), which is a perfect square. Recognizing this, we quickly know its square root is 10.
When we talk about simplifying expressions, noticing these perfect squares can save time and effort. In our example, after doing some multiplication and addition, we ended up with \(100\), which is a perfect square. Recognizing this, we quickly know its square root is 10.
- For instance, \(4^2 = 16\)
- Another example is \(7^2 = 49\)
Order of Operations
In mathematics, the order of operations is crucial for correctly evaluating expressions. We often learn this by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
First, address what is inside the parentheses - this is exactly what we did in our example by solving \(9 + 16\). The solution, 25, then becomes the focus of our next operation, which was multiplying by 4. \(4 \times 25 = 100\).
Lastly, the square root is computed, simplifying \(\sqrt{100}\) to \(10\).
First, address what is inside the parentheses - this is exactly what we did in our example by solving \(9 + 16\). The solution, 25, then becomes the focus of our next operation, which was multiplying by 4. \(4 \times 25 = 100\).
Lastly, the square root is computed, simplifying \(\sqrt{100}\) to \(10\).
- Always solve expressions in parentheses first.
- Next, handle any exponents or roots.
- And finally, perform multiplication, division, addition, and subtraction.
Other exercises in this chapter
Problem 7
For the following exercises, find the greatest common factor. \(200 p^{3} m^{3}-30 p^{2} m^{3}+40 m^{3}\)
View solution Problem 7
For the following exercises, identify the degree of the polynomial. \(-625 a^{8}+16 b^{4}\)
View solution Problem 7
For the following exercises, simplify the given expression. Write answers with positive exponents. \(3^{2} \times 3^{3}\)
View solution Problem 8
For the following exercises, simplify the rational expressions. \(\frac{m-12}{m^{2}-144}\)
View solution