Problem 137 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 534.7=5.347 \times 10^{3} $$

Step-by-Step Solution

Answer

The statement \(534.7=5.347 \times 10^{3}\) is false. The correct statement should be \(534.7=5.347 \times 10^{2}\).

Step 1: Converting to Standard Notation

To compare the two numbers, we must first convert the number written in scientific notation to standard notation. This implies multiplying \(5.347\) by \(10^{3}\), which equivalently means moving the decimal point three places to the right.

Step 2: Comparing

After converting to standard notation, we obtain \(5347\). Now, we can clearly see that \(5347\) is not equal to \(534.7\). The number \(5347\) is 10 times bigger than \(534.7\).

Step 3: Correct The Equation

To correct the equation, we should adjust the exponent in the scientific notation. The exponent should be \(2\) instead of \(3\). Thus, \(534.7 = 5.347 \times 10^{2}\) is the correct equation.

Key Concepts

Standard Notation

Standard notation is a way of writing numbers that is commonly used in everyday life. It presents numbers in their regular numeral form like 534.7 or 5347. In contrast, scientific notation is a shorthand way of writing very large or very small numbers using powers of ten.
If you're given a number in scientific notation, such as \(5.347 \times 10^{3}\), converting it to standard notation involves multiplying the base number (5.347) by the power of ten indicated by the exponent (\(10^3\)). This essentially means shifting the decimal point to the right 3 places, giving us 5347.

• Example: \(8.32 \times 10^2\) becomes 832 in standard notation.
• Example: \(7.4 \times 10^{-1}\) becomes 0.74 in standard notation.

Converting between these forms helps in understanding and comparing numbers easily.

Decimal Point

The decimal point is a crucial part of our numbering system, used to separate the whole number part of a number from its fractional part. When dealing with scientific notation, it's important to understand how moving the decimal point changes a number's value.
In the exercise, the scientific notation \(5.347 \times 10^{3}\) requires moving the decimal three places to the right, resulting in 5347. This movement increases the number significantly. Conversely, if the exponent is negative, you move the decimal to the left.

• Positive exponent: Move the decimal to the right.
• Negative exponent: Move the decimal to the left.

The position and movement of the decimal point determine the size of the number.

Exponents

Exponents represent repeated multiplication of a base number. In scientific notation, they show how many times you multiply the base number by ten.
An exponent of \(3\) in \(10^{3}\) tells us we multiply by ten three times, moving the decimal point three places to the right. This turns \(5.347\) into 5347, a much larger number.

• \(10^0\) means you multiply by 1 (no change).
• \(10^1\) means multiply by 10 (decimal moves one place).
• \(10^{-1}\) means divide by 10 (decimal moves left).

Understanding exponents is key to converting between scientific and standard notation and understanding their relationship.

Mathematical Equation Correction

Correcting mathematical equations involves checking each part for accuracy and making necessary adjustments. In our exercise, the original equation \(534.7 = 5.347 \times 10^{3}\) is found false after checking calculations.
This requires us to adjust the exponent to match the actual conversion to standard notation. The correct exponent, in this case, should be \(2\) instead of \(3\).

• Recalculation shows \(5.347 \times 10^{2}\) equates to 534.7, fixing the equation.
• Ensures both sides of the equation represent the same value once corrected.

Adjusting exponents appropriately helps reflect accurate mathematical expressions.