Question

For Exercises \(1-74,\) simplify. $$4 \sqrt{16 \cdot 9}-\left\\{(-4)^{3}+2[18 \div(-2)+(4-(-2))]\right\\}$$

Short answer

Question: Simplify the given expression and find its value: $$4\sqrt{16 \cdot 9} - \left\\{(-4)^{3}+2[18 \div(-2)+(4-(-2))]\right\\}$$. Answer: The simplified expression is $$118$$.

Step-by-step solution

Evaluate the expression inside the parentheses

First, evaluate the expression inside the parentheses, which are \((-4)^3\) and \(2[18 \div(-2)+(4-(-2))]\). Then apply the rules of operations of exponents, addition, and subtraction inside the parentheses.

Evaluate the expression inside the square root

Next, simplify the expression within the square root, \(16\cdot 9\). Multiply the two numbers and then find the square root of the result.

Evaluate the expression outside the parentheses

Now that we have the results inside the parentheses and the square root, we'll perform the remaining operations. Multiply 4 with the square root result and then subtract the result of the expression inside the parentheses. So, here is the step by step breakdown of the given expression: Step 1: Evaluate the expression inside the parentheses $$\left\\{(-4)^{3}+2[18 \div(-2)+(4-(-2))]\right\\} = (-4)^{3} + 2[(18 \div(-2))+ 6]$$ Step 2: Evaluate the expression inside the square root $$4 \sqrt{16 \cdot 9} = 4\sqrt{144} = 4 \times 12 = 48$$ Step 3: Evaluate the expression outside the parentheses $$(-4)^{3} = -64$$ $$18 \div(-2) = -9$$ $$-9 + 6 = -3$$ Now multiply by 2: $$-3 \cdot 2 = -6$$ Thus, the result of expression inside the parentheses is: $$-64 + (-6) = -70$$ Finally, subtract the result of the expression inside the parentheses from the square root result: $$48 - (-70) = 48 + 70 = 118$$ Therefore, the simplified expression is: $$118$$.

Key concepts

Order of Operations

To correctly simplify algebraic expressions, understanding the order of operations is crucial. Often remembered by the acronym PEMDAS, it stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It's a hierarchy used to avoid ambiguities in expressions.

For example, in the expression given in the exercise, one must first handle the calculations inside the parentheses, recognizing that exponents are evaluated before division and addition. Multiplications and divisions must also be treated equally, left to right, as they occur. This system is especially important when solving complex problems where different operations are combined, as it establishes a clear and unequivocal procedure for simplification. Short sentences and straightforward explanations, like these, can aid students in committing these rules to memory and applying them confidently.

Evaluating Square Roots

Evaluating square roots during simplification often confuses students, but it's simply a matter of finding a number that, when multiplied by itself, gives the original number under the square root sign, also known as a radicand.

Take, for instance, the square root of 144 in our exercise. The square root of 144 is 12, because 12 times 12 equals 144. Recognizing perfect squares like 144 and remembering common square roots can save time. However, not all square roots are integers, and in such cases, estimation or a calculator might be necessary. This step is essential before proceeding with any further operations, as it simplifies what otherwise might remain a bulky and intimidating expression.

Exponents in Algebra

Exponents in algebra represent how many times a number, known as the base, is multiplied by itself. In the given exercise, we see \( (-4)^3 \) which means \( -4 \) multiplied by itself three times. It's important to note the negative sign here — because it is raised to an odd power, the final answer remains negative.

One common pitfall for students is to incorrectly apply the negative sign when handling exponents. A negative base raised to an even exponent results in a positive number, while an odd exponent yields a negative result. Thoroughly comprehending how exponents work and paying attention to these details can prevent errors during the simplification of algebraic expressions containing exponents.