Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \frac{8 \times 10^{-3}}{5 \times 10^{-5}} $$

Short answer

The solution to the division \(\frac{8 \times 10^{-3}}{5 \times 10^{-5}}\) in scientific notation is \(1.6 \times 10^{2}\) and in decimal form is 160.

Step-by-step solution

Divide the Coefficients

Divide the numerical coefficients 8 and 5. The result is 1.6.

Subtract the Powers

The properties of exponents state that when dividing powers with the same base, you simply subtract the exponents. So subtract the exponent of the denominator from the exponent of the numerator: \(-3 - -5\). The result is \(10^{2}\).

Combine the Results

Combine the numerical part and the exponent part to get the answer in scientific notation: \(1.6 \times 10^2\).

Convert to Decimal Form

To convert it to decimal form, shift the decimal point in 1.6 two places to the right (as suggested by the positive power of 2), which gives us the decimal form: 160.

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in science and engineering to simplify measurements and calculations. In scientific notation, a number is written as the product of two parts: a coefficient and a power of ten. The coefficient must be a number greater than or equal to 1 and less than 10, and it's followed by the letter 'x' (or a multiplication sign) and then 10 raised to an exponent. For example, the number 300 can be written as 3 x 10² in scientific notation. This notation makes it easier to handle extremely large or small numbers, such as the distance between stars or the size of atoms.

When evaluating expressions, being familiar with scientific notation assists you in quickly simplifying and understanding the scale of the numbers involved.

Exponent Properties

Exponent properties, also known as the laws of exponents, are rules that describe how to handle mathematical expressions involving exponents. Here are some fundamental properties:\

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• Product of Powers: When multiplying two expressions with the same base, you add the exponents. So, \( x^a \times x^b = x^{a+b} \).
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• Quotient of Powers: Conversely, when dividing two expressions with the same base, subtract the exponents. This is \( \frac{x^a}{x^b} = x^{a-b} \).
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• Power of a Power: When raising an exponential expression to another exponent, you multiply the exponents. That's \( (x^a)^b = x^{a \times b} \).
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\Understanding and correctly applying these properties are crucial in simplifying expressions and performing precise calculations in algebra. They especially come into play when working with scientific notation, as both involve the manipulation of powers of ten.

Dividing Coefficients

Dividing coefficients is an elementary yet vital operation in algebra, particularly when working with expressions in scientific notation. Coefficients are the numerical parts of terms in an algebraic expression. When you divide one coefficient by another, you perform standard division. For instance, with \( \frac{8}{5} \), you simply divide 8 by 5 resulting in 1.6. This new coefficient is then often paired with an exponent to complete an expression in scientific notation.

It's important to keep the resulting coefficient between 1 and 10 when formulating scientific notation. If not, you'd need to adjust the coefficient and exponent accordingly to maintain proper notation. A well-understood method of dividing coefficients aids in simplifying complex problems into manageable calculations.

Converting to Decimal Form

Converting an expression from scientific notation to decimal form is a necessary skill for interpreting real-world data and for understanding the magnitude of results in disciplines such as physics and chemistry. This conversion requires you to manipulate the position of the decimal point in the coefficient, depending on the power of ten. If the exponent is positive, move the decimal point to the right; if negative, move it to the left. The number of places you move the decimal is equivalent to the value of the exponent.

For example, for \(1.6 \times 10^2\), the exponent is 2, which means the decimal point in 1.6 needs to be moved two places to the right, resulting in 160. Mastering the skill of converting scientific notation to decimal form allows students to visualize numbers in a more conventional and familiar format, thereby making it easier to understand the magnitude of such numbers in a real-world context.