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Chapter 1: Problem 25

Evaluate each radical. $$\sqrt{3+13}+9$$

Short Answer

Expert verified
The result is 13.

Step by step solution

01

- Evaluate the expression inside the radical

First, calculate the sum inside the square root. Add 3 and 13. \[ 3 + 13 = 16 \]
02

- Simplify the radical expression

Next, find the square root of the result from Step 1. \[ \sqrt{16} = 4 \]
03

- Add the outer term

Finally, add 9 to the simplified radical value obtained in Step 2. \[ 4 + 9 = 13 \]

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Key Concepts

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

simplifying radicals
Radicals can often seem complex, but simplifying them makes them much easier to handle. The key to simplification is to find the prime factors of the number inside the radical.
For example, let's consider the radical expression \(\sqrt{16} \).
Here are the steps to simplify it:
  • Find the prime factors of 16, which are 2 x 2 x 2 x 2.
  • Group the prime factors into pairs: (2 x 2) and (2 x 2).
  • Take one number from each pair out of the radical: 2 x 2 = 4.
So, \(\sqrt{16} \) simplifies to 4.
By breaking down the number inside the radical, the expression becomes less intimidating. This method can be applied to any radical.
square roots
Square roots are a common type of radical expression. They help in finding the original number when multiplied by itself gives the input value.
For example, \(\sqrt{16} \) means finding a number that, when multiplied by itself, gives 16.
Here are some fundamental properties of square roots:
  • The square root of a perfect square (like 4, 9, or 16) is always an integer.
  • \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \): The product property.
  • \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \): The quotient property.
If dealing with non-perfect squares, we can approximate or leave the square root in its radical form. Always remember, simplifying square roots requires looking for perfect square factors.
addition in algebra
Addition is one of the basic operations in algebra, and it鈥檚 crucial for simplifying expressions and solving equations.
When evaluating expressions involving addition, as seen in \(\sqrt{3+13}+9 \), you should follow these steps:
  • First, perform the addition inside the radical.
  • Simplify the resulting radical expression.
  • Finally, add the outer term to the simplified radical value.
In our example, we calculated inside the radical first: \(3 + 13 = 16 \). Then, we simplified the radical: \(\sqrt{16} = 4 \). Finally, we added the outer term: \(4 + 9 = 13 \).
Practicing these steps will help you confidently handle more complex algebraic expressions involving addition.

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Most popular questions from this chapter

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