Problem 40 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{10^{4}}{10^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 10^2 \).

Step 1: Identify the Exponents

The given expression is \( \frac{10^4}{10^2} \). Here, we need to simplify this expression using the quotient rule for exponents. Identify that the numerator has an exponent of 4 and the denominator has an exponent of 2.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Apply this rule to our expression: \( \frac{10^4}{10^2} = 10^{4-2} \).

Step 3: Simplify the Expression

Calculate the result of the exponent subtraction: \( 4 - 2 = 2 \). Therefore, \( \frac{10^4}{10^2} = 10^2 \).

Step 4: Present the Final Simplified Expression

Thus, the expression simplifies to \( 10^2 \).

Key Concepts

Quotient Rule

The quotient rule is a fundamental principle when working with exponents. It helps simplify expressions that involve dividing two powers with the same base. When you see a division like \( \frac{a^m}{a^n} \), you should remember this handy rule:

• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)

This means you subtract the exponent in the denominator from the exponent in the numerator. For example, if you have \( \frac{10^4}{10^2} \), according to the quotient rule, you would do \( 10^{4-2} \), which simplifies to \( 10^2 \). This shortcut allows you to more easily manage expressions with exponents.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing an expression to its simplest form. When it comes to expressions with exponents, using the right rules makes this process much easier. After applying the quotient rule to our division of powers, the next step is just to "simplify" the expression using basic arithmetic.

• Apply relevant rules (like the quotient rule)
• Perform any simple arithmetic operations
• Rewrite the expression as simply as possible

For example, once you apply the quotient rule to \( \frac{10^4}{10^2} \) and get \( 10^{4-2} \), you simply perform the subtraction to end up with \( 10^2 \). This simpler form is easier to work with in computations.

Power Subtraction

Power subtraction is a critical step in simplifying expressions with exponents. It directly follows from the quotient rule. Once you align a division of powers, the core task is subtracting the exponent in the denominator from the exponent in the numerator.Consider the expression \( \frac{10^4}{10^2} \). After applying the quotient rule, you are left with \( 10^{4-2} \). To fully simplify it, you subtract the exponents:

• Numerator exponent: 4
• Denominator exponent: 2
• Subtraction: \( 4 - 2 = 2 \)

So, \( 10^{4-2} \) becomes \( 10^2 \). This power subtraction is straightforward yet crucial for simplifying expressions effectively.