Question

Use a calculator to evaluate the expression. Round your answer to two decimal places. $$(-6.3)^{2}(9.5)(4.8)$$

Short answer

The evaluation of the expression (-6.3)^2*(9.5)*(4.8) equals 1809.86 when rounded to two decimal places.

Step-by-step solution

Evaluate the Power

Before executing the multiplication, evaluate the power of -6.3. The expression (-6.3)^2 means -6.3 multiply by itself. So, (-6.3) * (-6.3) = 39.69

Evaluate the Products

Next, multiply the result from Step 1 by 9.5 and then by 4.8. Therefore, 39.69 * 9.5 = 377.055 and then 377.055 * 4.8 = 1809.864.

Round to Two Decimal Places

Lastly, round off the results from Step 2 to two decimal places. So, the final result will be 1809.86.

Key concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent tells us how many times to multiply the base by itself. For example, in the expression \( (-6.3)^2 \), the base is -6.3 and the exponent is 2. This means you need to multiply -6.3 by itself exactly once. The process seems simple when the exponent is a small integer, but it becomes more complex with larger or fractional exponents.

Calculators are quite handy for exponentiation. They have a specific button, often labeled as '^' or sometimes as 'exp', that allows you to input the base and the exponent. In our example, entering -6.3 and then pressing '^' followed by 2 will give the result 39.69. Always remember to enclose negative bases in parentheses to ensure the calculation is accurate, as the negative sign has a significant impact on the result.

Handling Negative Bases

When the base is negative and the exponent is an even number, the result will always be positive. This is because multiplying a negative number by itself an even number of times will result in a positive number. Conversely, with an odd exponent, the result will remain negative. It's essential to understand this concept to correctly evaluate expressions involving exponentiation.

Rounding Decimals

Rounding decimals is a way of simplifying a number while trying to keep its value similar to what it was. It's particularly useful when you want to make a long decimal number easier to work with or communicate. When you round a number, you look at the digit to the right of the place you're rounding to determine whether to round up or down.

For instance, the problem requires the answer to be rounded to two decimal places. Here's a quick guide:

• If the third decimal digit is 5 or higher, you increase the second decimal digit by one. This is known as rounding up.
• If the third decimal digit is 4 or lower, you leave the second decimal digit as it is. This is called rounding down.

Let's apply this to our exercise. We have the number 1809.864, and we're rounding to two decimal places. The third decimal place is a 4, so according to the rule, we round down, keeping the second decimal place as is. The final rounded number is 1809.86. Rounding to the correct decimal place is crucial, especially in fields like finance and engineering, where precision matters greatly.

Order of Operations

Understanding the order of operations is crucial when evaluating mathematical expressions. It ensures that everyone solves the expression the same way, without ambiguity. The most common rule is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This order tells you to first calculate expressions within parentheses, then perform exponentiation, followed by multiplication and division in the order they appear from left to right, and finally, carry out addition and subtraction from left to right.

Applying Order of Operations

In our example, the first step according to PEMDAS is to evaluate the exponentiation \( (-6.3)^2 \). This gives us 39.69. As there are no parentheses to consider next, we move on to multiplication. We multiply 39.69 first by 9.5 and then by 4.8, maintaining the left-to-right rule for multiple multiplications. There's no addition or subtraction here, so we're done with the calculations after the multiplications. Remember that following the proper order is key to obtaining the correct result in any mathematical evaluation.