Question

For Exercises \(1-74,\) simplify. $$-5 \sqrt{36}+40 \div(-5) \cdot 2-14$$

Short answer

Answer: The simplified form of the expression is -60.

Step-by-step solution

Simplify the Square Root

Find the square root of 36: $$\sqrt{36} = 6$$ Now the expression can be rewritten as: $$-5(6) + 40 \div(-5) \cdot 2 - 14$$

Perform Multiplication and Division

First, we will multiply -5 by 6: $$-5(6) = -30$$ Next, we will divide 40 by -5: $$40 \div(-5) = -8$$ Now, multiply -8 by 2: $$-8 \cdot 2 = -16$$ Finally, substitute these results back into the expression: $$-30 + (-16) - 14$$

Simplify the Expression

Add -30 and -16: $$-30 + (-16) = -46$$ Lastly, subtract 14 from -46: $$-46 - 14 = -60$$ The simplified expression is: $$-60$$

Key concepts

Square Root Simplification

Understanding square root simplification is critical when simplifying algebraic expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. Simplifying square roots involves finding the prime factors of the number and pairing them to pull them out of the radical. For instance, the square root of 36 is indeed 6 because 6 multiplied by itself gives us 36, which completes the simplification process for this part of any expression.

When you are simplifying expressions that involve square roots, your goal is to find the simplest radical form. Sometimes, the given number might not be a perfect square like 36, which makes its square root an integer. In such cases, you should look for the largest square factor of the number to simplify the expression partially. For example, for \(\sqrt{50}\), you would recognize that 50 is 25 times 2, and since 25 is a perfect square, you can simplify this to \(5\sqrt{2}\), with 5 being outside of the radical because it's the square root of 25.

Order of Operations

The order of operations is essential to correctly simplify algebraic expressions. It is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This rule is crucial because performing arithmetic operations in an incorrect order can lead to the wrong answer.

In the given exercise, we first simplified the square root before performing any other operations. After that, we followed the order of operations by dealing with multiplication and division first, then addition and subtraction. To illustrate, after simplifying the square root to 6, the expression \( -5 \sqrt{36}+40 \div(-5) \cdot 2 -14 \) was correctly addressed by first multiplying -5 by 6, then dividing 40 by -5, and following up with multiplication of the quotient by 2, before finally moving on to addition and subtraction.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks of most algebraic expressions. Mastery of these operations is essential when solving mathematical expressions. In our exercise, after simplifying the square root, the remaining steps involve these fundamental operations.

We've performed multiplication, getting \( -5 \cdot 6 = -30 \), and division, \( 40 \div(-5) = -8 \). Then, after multiplying the division's result by 2 (\( -8 \cdot 2 = -16 \)), we were left with a simpler expression that only required addition and subtraction to finalize the simplification. Remember to always multiply or divide before proceeding to addition or subtraction. The final arithmetic operation was adding and subtracting the resulting numbers to reach the simplified expression of \( -60 \).