Problem 34 - Elementary Algebra

Question

\(\frac{146 \times 10^{8}}{2 \times 10^{-5}}\)

Step-by-Step Solution

Answer

73 \times 10^{13}

Step 1: Identify the components of the expression

The expression is \(\frac{146 \times 10^{8}}{2 \times 10^{-5}}\). We have 146 and \(10^{8}\) in the numerator, and 2 and \(10^{-5}\) in the denominator.

Step 2: Simplify the coefficients

Divide the coefficients: \( \frac{146}{2} = 73\).

Step 3: Simplify the exponents

Using the properties of exponents, divide the exponents: \( \frac{10^{8}}{10^{-5}} = 10^{8 - (-5)} = 10^{8 + 5} = 10^{13}\).

Step 4: Combine the simplified components

Putting it all together, we get: \( 73 \times 10^{13}\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with scientific notation and similar concepts. One key rule we often use is that when dividing two exponential terms with the same base, we subtract the exponent in the denominator from the exponent in the numerator. In the exercise, this rule is applied as follows: ewline ewline ewline Let's break it down:

• We start with \(\frac{10^8}{10^{-5}}\), which means we need to divide the terms.
• Using the rule, we subtract -5 (the exponent in the denominator) from 8 (the exponent in the numerator).
• Subtracting a negative is the same as adding: 8 - (-5) = 8 + 5.
• This gives us a new exponent of 13, so \[10^8 - (-5) = 10^{8+5} = 10^{13}.\]

Rocket science? Nope! Just good ol' subtraction with a bit of a twist from the exponents.

Simplifying Coefficients

Simplifying coefficients is just a fancy term for dividing the obvious numbers in our expression. These steps remind us that math can be straightforward and logical. In the provided exercise: ewline ewline ewline Let's see how we did this:

• Look at the coefficients in the numerator and denominator: 146 in the numerator and 2 in the denominator.
• When we divide these, it is just basic division: \(\frac{146}{2}= 73.\)

And there it is, our simplified coefficient is 73. No surprises here, just divide as you would in any normal situation.

Division of Powers of Ten

The division of powers of ten might look complex, but it’s essentially applying exponent rules with the base of ten. Here's a closer look at this part from the exercise: ewline ewline ewline Here's how it goes:

• Starting with \(10^8\) in the numerator and \(10^{-5}\) in the denominator.
• According to exponent rules, you subtract the exponent in the denominator from the exponent in the numerator: \[10^{8 - (-5)}.\]
• This boils down to adding 5 to 8 because subtracting a negative is adding: \(8 + 5 = 13.\)
• Simplifying further, we get \(10^{13}.\)

In a nutshell, when dividing powers of ten, remember to subtract the exponents (or add if you're subtracting a negative). This keeps the process simple and the results accurate.