Question

Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation. a) \(456 \times\left(7.4 \times 10^{8}\right)=?\) b) \(\left(3.02 \times 10^{5}\right) \div\left(9.04 \times 10^{15}\right)=?\) c) \(0.0044 \times 0.000833=?\)

Short answer

a) \(3.3644 \times 10^{11}\); b) \(3.34 \times 10^{-11}\); c) \(3.6652 \times 10^{-6}\).

Step-by-step solution

Multiply 456 by 7.4

Start with the expression from part a).First, calculate the product of the numbers: \( 456 \times 7.4 \).Using a calculator, you will find that:\( 456 \times 7.4 = 3364.4 \).

Multiply by Power of Ten

Next, incorporate the power of ten from part a).Multiply the result from Step 1 by \( 10^8 \).\( 3364.4 \times 10^8 = 3.3644 \times 10^{11} \).

Express Result in Scientific Notation - Part a

The expression \( 3364.4 \times 10^8 \) is already fairly close, but it should be represented as \( 3.3644 \times 10^{11} \), ensuring the number is less than 10.

Divide Numbers for Part b

Move on to part b) and solve \( \frac{3.02 \times 10^5}{9.04 \times 10^{15}} \).Calculate \( 3.02 \div 9.04 \) using a calculator, resulting in:\( 3.34 \times 10^{-1} \approx 0.334 \).

Subtract Powers of Ten

Using the quotient from Step 4, incorporate the powers of ten subtraction:\( 10^5 \div 10^{15} = 10^{5-15} = 10^{-10} \).Combine the results: \( 0.334 \times 10^{-10} \).

Express Result in Scientific Notation - Part b

For a proper scientific notation, express it as:\( 3.34 \times 10^{-11} \).

Multiply Decimal Numbers for Part c

Calculate the product from part c) using the numbers given:\( 0.0044 \times 0.000833 \).Using a calculator to multiply these, you get:\( 3.6652 \times 10^{-6} \).

Express Result in Scientific Notation - Part c

Convert the calculated product into scientific notation by expressing \( 3.6652 \times 10^{-6} \) where the number is in the form \( a \times 10^n \).

Key concepts

Calculator Methods

Calculators are essential tools when solving expressions in scientific notation. Understanding how to use them efficiently can simplify calculations like multiplication and division with large or small numbers. When entering numbers in scientific notation, remember to input them in sections. Specifically, you enter the decimal part first, followed by the 'E' or 'EXP' button, which represents the power of ten, and finally the exponent itself. For instance, entering \(7.4 \times 10^{8}\) on most calculators involves inputting '7.4', pressing the 'E' button, and then typing '8'. This method ensures you don't have to manually calculate large powers or decimal places.
When performing operations such as in problem a), first multiply the decimal components (e.g., 456 and 7.4) to get an intermediate result. Then, apply the power of ten separately if your calculator does not handle scientific notation directly. This systematic approach avoids potential errors.
For division problems like part b), the process involves similar steps: divide the coefficients first (e.g., 3.02 divided by 9.04), and then deal with the exponents separately. Combining these results gives you the final answer in scientific notation.

Power of Ten

Understanding the concept of "Power of Ten" is crucial in scientific notation. This notation format allows us to express very large or small numbers in a more manageable form. Each power of ten is a representation of how many times a number is multiplied by ten. For instance, in the expression \(10^8\), the number 8 is the exponent, indicating 10 is multiplied by itself 8 times, equating to 100,000,000.
When solving expressions in scientific notation, the power of ten plays a central role. If given \(456 \times (7.4 \times 10^{8})\), understanding that \(10^8\) represents a hugely multiplied number is key. In part b), division involves subtracting exponents because dividing powers reduces the exponent count. Thus, \(10^5 \div 10^{15}\) becomes \(10^{5-15}\), simplifying to \(10^{-10}\).
The power of ten easily adjusts the scale of a number, making it straightforward to understand and manipulate extremely large or small values without cumbersome zeroes.

Multiplication and Division in Scientific Notation

Multiplication and division in scientific notation involve straightforward rules that simplify working with powers of ten. When multiplying two numbers, you multiply their coefficients and add the exponents. For example, to solve part a) \((456 \times 7.4) \times 10^8\), you first multiply the smaller numbers (456 with 7.4) to obtain 3364.4. Next, the exponent of ten (in this case, \(10^8\)) stays the same or increases if additional powers are involved.
For division, as seen in part b), you divide the coefficients and subtract the exponents. If dividing \(3.02 \times 10^5\) by \(9.04 \times 10^{15}\), you first divide 3.02 by 9.04, getting approximately 0.334. Simultaneously, subtract the exponents (5 minus 15), resulting in \(-10\), and combine with the coefficient for a solution in scientific notation \(3.34 \times 10^{-11}\).
These operations often require expressing the final number in 'proper' scientific notation, usually by ensuring the coefficient is a number between 1 and 10. This format helps standardize results and maintain clarity throughout calculations.