Create and solve two problems involving addition, subtraction, multiplication, and division of radicals. The two problems combined should contain all four operations. For example, one problem could combine addition and multiplication and the second problem could contain subtraction and division. Check out the following sample problem to help you get started.
Sure, I can create two problems involving addition, subtraction, multiplication, and division of radicals for you.
Problem 1: Addition and Multiplication
Solve: √12 + 3√27 + 2√48
To solve this problem, we need to simplify each radical first and then add them together.
√12 = √(4 * 3) = 2√3
√27 = √(3 * 9) = 3√3
√48 = √(16 * 3) = 4√3
Now, substitute these values back into the original problem:
2√3 + 3√3 + 4√3
The radicals have the same value, which is √3. So, we can combine them:
2√3 + 3√3 + 4√3 = (2 + 3 + 4)√3 = 9√3
Therefore, the solution to the problem is 9√3.
Problem 2: Subtraction and Division
Solve: √60 - 4√15 ÷ √3
To solve this problem, we need to simplify each radical first and then perform the operations.
√60 = √(4 * 15) = 2√15
√15 cannot be simplified further.
Now, substitute these values back into the original problem:
2√15 - 4√15 ÷ √3
Since we have division in this problem, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of √3, which is √3:
(2√15 - 4√15) * √3 / (√3 * √3)
Simplify the expression:
(2√15 - 4√15) * √3 / 3
Combine like terms:
(-2√15) * √3 / 3
Now, multiply:
-2√45 / 3
Since √45 = √(9 * 5) = 3√5, we can substitute the value:
-2 * 3√5 / 3
Cancel out common factors of 3:
-2√5
Therefore, the solution to the problem is -2√5.
I hope this helps you understand how to solve problems involving addition, subtraction, multiplication, and division of radicals!
Problem 1:
Solve the following expression: $\sqrt{16} + \sqrt{9} \cdot \sqrt{4} - \sqrt{25} \div \sqrt{5}$.
Solution:
First, let's simplify each radical term individually:
$\sqrt{16} = 4$, since 4 squared is 16,
$\sqrt{9} \cdot \sqrt{4} = 3 \cdot 2 = 6$, and
$\sqrt{25} \div \sqrt{5} = 5 \div \sqrt{5}$.
Now, we can simplify the expression further:
$4 + 6 - 5 \div \sqrt{5}$,
$4 + 6 - \frac{5}{\sqrt{5}}$.
Next, rationalize the denominator of the fraction by multiplying the numerator and denominator by $\sqrt{5}$:
$4 + 6 - \frac{5 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}$,
$4 + 6 - \frac{5 \cdot \sqrt{5}}{5}$.
Simplify the expression:
$4 + 6 - \sqrt{5}$,
$10 - \sqrt{5}$.
Therefore, the solution to the expression $\sqrt{16} + \sqrt{9} \cdot \sqrt{4} - \sqrt{25} \div \sqrt{5}$ is $10 - \sqrt{5}$.
Problem 2:
Solve the following equation: $2 \cdot \sqrt{9} - (\sqrt{25} - 1) \div \sqrt{2}$.
Solution:
First, simplify each radical term individually:
$\sqrt{9} = 3$, since 3 squared is 9,
$\sqrt{25} = 5$, and
$\sqrt{2}$ remains the same.
Now, we can solve the equation further:
$2 \cdot 3 - (5 - 1) \div \sqrt{2}$,
$6 - 4 \div \sqrt{2}$.
Next, divide 4 by $\sqrt{2}$:
$6 - \frac{4}{\sqrt{2}}$,
$6 - \frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}$,
$6 - \frac{4 \cdot \sqrt{2}}{2}$,
$6 - 2 \cdot \sqrt{2}$.
Therefore, the solution to the equation $2 \cdot \sqrt{9} - (\sqrt{25} - 1) \div \sqrt{2}$ is $6 - 2 \cdot \sqrt{2}$.
Sure! Here are two problems involving addition, subtraction, multiplication, and division of radicals:
Problem 1: Addition and Multiplication
Simplify and compute the value of:
√12 + √3 * √4
Solution:
First, simplify each radical:
√12 = √(4 * 3) = √4 * √3 = 2√3
√4 = 2
Now substitute the simplified values into the original expression:
2√3 + √3 * 2
Apply the distributive property:
2√3 + 2√3
Combine like terms:
4√3
Answer: 4√3
Problem 2: Subtraction and Division
Simplify and compute the value of:
(√20 - √5) / (√2)
Solution:
First, simplify each radical:
√20 = √(4 * 5) = √4 * √5 = 2√5
√5 = √(5)
Now substitute the simplified values into the original expression:
(2√5 - √5) / (√2)
Combine like terms in the numerator:
(2√5 - √5) / (√2) = √5 / (√2)
Rationalize the denominator by multiplying both the numerator and denominator by √2:
(√5 * √2) / (√2 * √2)
simplify:
√10 / 2
Answer: (√10) / 2
I hope this helps! Let me know if you have any further questions.