Question

Simplify the expression. Then use a calculator to evaluate the expression. Round the result to the nearest tenth when appropriate. $$ (5.0 \cdot 4.9)^{2} $$

Short answer

The simplified expression is \(24.5^2\) and after calculating and rounding, the result is 600.3.

Step-by-step solution

Simplify the Expression

First we need to simplify the expression within the brackets. This means we multiply \(5.0\) by \(4.9\).

Evaluate the Squared Term

After multiplying, take the result and square it according to the exponent in the expression. Squaring a number means multiplying it by itself.

Round the Result

After evaluating the squared term, use a calculator to get the answer. Finally, round the result to the nearest tenth as per the instructions of the problem.

Key concepts

Order of Operations

Understanding the order of operations is crucial for simplifying algebraic expressions correctly. This set of rules dictates the sequence in which you should perform mathematical operations to achieve the correct result. The commonly used acronym is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, we are asked to simplify \( (5.0 \cdot 4.9)^{2} \). According to PEMDAS, we must deal with the operation inside the parentheses first—that is, the multiplication of 5.0 by 4.9. Once this is completed, we then proceed to the next operation indicated by the exponent: squaring the result of the multiplication.

Here's the trick: always remember to follow the order of operations step by step to avoid any mistakes. If you encounter additional operations such as square roots or factorial in more complex expressions, consult the extended rules and apply them accordingly.

Exponents and Powers

The concept of exponents and powers can initially seem daunting, but it's all about understanding that an exponent indicates how many times to multiply a number by itself. In the expression \( (5.0 \cdot 4.9)^{2} \), the exponent is 2, indicating that we must multiply the result of the expression within the parentheses by itself once.

To clarify, squaring a number (using an exponent of 2) simply means multiplying the number by itself: \( a^2 = a \cdot a \). For the exercise, once you've multiplied 5.0 by 4.9, you then take that result and square it. It's important to note that exponents are not limited to squares; there are cubes (exponent of 3), and so on, all following the same principle. It's this powerful shorthand notation that allows us to perform multiplication of the same number several times in a much simpler way.

Rounding Decimals

Rounding decimals is a key skill in ensuring answers are practical, especially in real-life applications where precision to numerous decimal places isn't always necessary or useful. The rule of thumb is to look at the number in the place immediately to the right of the place you are rounding to. If this number is 5 or greater, you round up. If it's less than 5, you keep the digit the same and remove all the numbers after it.

For instance, when the exercise requires rounding to the nearest tenth, you focus on the first digit after the decimal point. If the squared result of the expression \( (5.0 \cdot 4.9)^{2} \) is, for example, 120.29, you would look at the second decimal place which is 9. Since 9 is greater than 5, you would round the number in the tenths place up, resulting in 120.3. The importance of mastering rounding decimals cannot be overstated, as it is a frequently used technique in both academic problems and everyday life.