Question

For Exercises \(1-74,\) simplify. $$(-2)^{3}+3 \sqrt{25}-18 \div 3$$

Short answer

Question: Simplify the given expression and find its value: \((-2)^3 + 3\sqrt{25} - 18 \div 3\) Answer: The value of the given expression is 1.

Step-by-step solution

Exponentiation

In the first part of the expression, we have \((-2)^3\). Raise the base \(-2\) to the power of \(3\): $$(-2)^3 = (-2) \cdot (-2) \cdot (-2) = 4 \cdot (-2) = -8$$

Square Root

In the second part, we need to calculate the square root of \(25\): $$\sqrt{25} = 5$$

Division

In the third part, we have the division \(18 \div 3\). Here, we need to calculate the result of \(18\) divided by \(3\): $$18 \div 3 = 6$$

Simplify and add the values

Now that we have calculated the values for each part of the expression, we can add them together: $$-8 + 3(5) - 6$$ Multiply \(3(5)\) to get \(15\) and add it to \(-8\): $$-8 + 15 = 7$$ Finally, subtract the value from the division, which was \(6\): $$7 - 6 = 1$$ The resulting value of the given expression is \(1\).

Key concepts

Exponentiation

Exponentiation is a fundamental mathematical operation involving two numbers: the base and the exponent. When you raise a base to a power, you multiply the base by itself as many times as the exponent indicates. For example, \( -2^3 \) means you multiply \( -2 \) by itself three times: \( (-2) \cdot (-2) \cdot (-2) = 4 \cdot (-2) = -8 \).

Remember, when the base is negative and the exponent is odd, the result will be negative, as seen in the exercise given. If the exponent is even, the result would be positive due to the fact that the product of two negative numbers is positive. This concept is vital for simplifying expressions and forms the foundation of more complex mathematical functions.

Square Root Calculation

Calculating square roots is another essential concept in simplifying mathematical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of 25 is 5, because \( 5 \times 5 = 25 \).

The symbol for the square root is \( \sqrt{} \), and finding the square root is the inverse operation of squaring a number. It's important to note that every positive number has two square roots: a positive and a negative root, but typically, when we refer to 'the square root,' we mean the positive one. In the context of our exercise, \( \sqrt{25} \), we are referring to the positive root, which is 5.

Arithmetic Division

Arithmetic division is a basic arithmetic operation where we divide one quantity by another. It answers questions such as 'how many times does this number fit into that number?' In the given exercise, we need to find out how many times 3 fits into 18, which is \( 18 \div 3 = 6 \).

It's essential to distinguish between division and subtraction; division is used when we want to know about equal parts or shares, whereas subtraction is used to find out how much remains when a portion is taken away. When simplifying expressions, performing division correctly allows us to reduce terms and combine like terms effectively.

Order of Operations

Understanding the order of operations is critical when simplifying expressions to ensure we perform calculations correctly. The order, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction), dictates which calculations to carry out first.

In expressions such as \( (-2)^{3}+3 \sqrt{25}-18 \div 3 \), we must first deal with the exponentiation (\( (-2)^{3} \)), then tackle any roots or powers (\( \sqrt{25} \)), followed by multiplication or division (\( 18 \div 3 \)), and, only afterwards, handle any addition or subtraction. Following these steps ensures a correct and universally accepted solution.