Problem 10 - Elementary and Intermediate Algebra

Question

Simplify each expression. a. \(10^{24} \times 10^{33}\) b. \(\frac{10^{50}}{10^{36}}\) c. \(\frac{10^{15} \times 10^{27}}{10^{40}}\)

Step-by-Step Solution

Answer

a. \(10^{57}\); b. \(10^{14}\); c. \(10^{2}\).

Step 1: Understanding Exponent Multiplication Rule

When multiplying two numbers with the same base, we add their exponents. For example, when multiplying \( 10^a \) and \( 10^b \), the result is \( 10^{a+b} \).

Step 2: Applying Exponent Multiplication Rule to Part a

For the expression \( 10^{24} \times 10^{33} \), we add the exponents: \( 24 + 33 = 57 \). Thus, \( 10^{24} \times 10^{33} = 10^{57} \).

Step 3: Understanding Exponent Division Rule

When dividing two numbers with the same base, we subtract the exponent of the denominator from the exponent of the numerator. For example, \( \frac{10^a}{10^b} \) results in \( 10^{a-b} \).

Step 4: Applying Exponent Division Rule to Part b

For \( \frac{10^{50}}{10^{36}} \), subtract the exponents: \( 50 - 36 = 14 \). Therefore, \( \frac{10^{50}}{10^{36}} = 10^{14} \).

Step 5: Simplifying Complex Exponent Expressions

For expressions involving both multiplication and division, first perform multiplication in the numerator, then apply the division rule by subtracting the total numerator exponent by the denominator exponent.

Step 6: Simplifying Part c Numerator with Multiplication Rule

Multiply the exponents in the numerator: \( 10^{15} \times 10^{27} \) by adding the exponents: \( 15 + 27 = 42 \). So, \( 10^{15} \times 10^{27} = 10^{42} \).

Step 7: Simplifying Part c with Division Rule

Use the result from Step 6 in \( \frac{10^{42}}{10^{40}} \) and subtract the exponents: \( 42 - 40 = 2 \). Hence, \( \frac{10^{15} \times 10^{27}}{10^{40}} = 10^{2} \).

Key Concepts

Exponent Multiplication Rule

When dealing with exponents, particularly in multiplication, the Exponent Multiplication Rule is essential and can make complex calculations a lot simpler. This rule states that when you multiply two powers with the same base, you simply add the exponents together while keeping the base unchanged.
For example, if you have expressions like

• \(10^{24} \times 10^{33}\),

you add the exponents (in this case, 24 and 33) to get the result:

• 
  \(10^{24+33} = 10^{57}\).

This rule makes it clear that instead of multiplying large numbers directly, you can simplify the process by working with their exponents.
Remember, this only works when the bases

• are the same;
• otherwise, different rules apply.

But with widely used bases like 10, this rule can help streamline many calculations into easier to manage expressions.

Exponent Division Rule

The Exponent Division Rule handles situations when you're dividing numbers that have exponents with the same base. Using this rule, you subtract the exponent in the denominator from the exponent in the numerator. This simplifies the division of numbers with powers by avoiding direct computation with potentially large numbers. For instance, consider:

• \(\frac{10^{50}}{10^{36}}\).

According to the division rule, you subtract 36 from 50:

• \(10^{50-36} = 10^{14}\).

This trick works because it's rooted in the idea that dividing by a number is the same as multiplying by its reciprocal, which in terms of powers, means reducing the exponent.
This method is incredibly efficient for working with very large or very small numbers, especially in scientific and engineering fields where such notations are frequent.

Simplifying Exponential Expressions

Simplifying expressions with exponents can often involve both multiplication and division tasks. In such cases, applying both the Exponent Multiplication and Division Rules systematically simplifies the expressions to their core form.
Here's the approach for an example like:

• \(\frac{10^{15} \times 10^{27}}{10^{40}}\).

First, solve the numerator by applying the Multiplication Rule:

• \(10^{15+27} = 10^{42}\).

Next, apply the Division Rule to simplify the fraction:

• \(\frac{10^{42}}{10^{40}} = 10^{42-40} = 10^{2}\).

The combination of these steps reduces the potentially cumbersome exponential expression into a much simpler form, making it easier to interpret and use.
These rules offer powerful shortcuts for those working with exponential numbers, ensuring that we can manage and operate them in a straightforward and efficient manner without compromising on accuracy.