Problem 11 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \frac{10^{7}}{10^{4}} $$

Step-by-Step Solution

Answer

The evaluated expression is 1000.

Step 1: Applying the Quotient of Powers Property

To evaluate the expression \( \frac{10^{7}}{10^{4}} \), we can apply the quotient of powers property. This states that \( \frac{a^m}{a^n} = a^{m-n} \), provided \( a eq 0 \). Here, \( a = 10 \), \( m = 7 \), and \( n = 4 \).

Step 2: Simplifying the Expression

Now, apply the exponents' subtraction: \( 10^{7-4} = 10^{3} \). This simplifies the expression to \( 10^{3} \).

Step 3: Calculate the Power of Ten

The expression \( 10^{3} \) means \( 10 \) raised to the power of \( 3 \). Calculate this by multiplying three tens together: \( 10 \times 10 \times 10 = 1000 \). Thus, \( 10^{3} = 1000 \).

Key Concepts

Simplifying Exponential Expressions

Simplifying exponential expressions can be much simpler if you use certain mathematical properties. One of the most useful for dealing with expressions that have the same base is the "Quotient of Powers Property". This property tells us that when we divide two powers with the same base, we can subtract the exponents:

• If you have an expression like \( \frac{a^m}{a^n} \), you can simplify it to \( a^{m-n} \).
• Here, the base \( a \) must not be zero, and \( m \) and \( n \) are the exponents.

In this exercise, we have \( \frac{10^7}{10^4} \), which simplifies to \( 10^{7-4} \). Hence, the expression reduces to \( 10^3 \). This illustrates how exponent properties simplify potentially complex expressions into manageable calculations.

Powers of Ten

The power of ten is an exciting and essential concept, especially when dealing with large numbers or simplifying expressions. When you see \( 10^3 \) or similar expressions, you're working with powers of ten, which follow some clear principles:

• In an expression like \( 10^n \), the 10 is the base, and \( n \) is the exponent.
• It represents multiplying 10 by itself \( n \) times. So, \( 10^3 \) means \( 10 \times 10 \times 10 \).

Powers of ten are especially popular because they simplify arithmetic in base-10, which is the numbering system we use daily. In this problem, the final answer \( 10^3 = 1000 \), is easy to verify by multiplying three tens in a row.

Exponent Subtraction

Exponent subtraction is crucial when simplifying fractions of exponential expressions with the same base. Using exponent subtraction, we follow these steps:

• When dividing two exponential expressions, subtract the exponent of the denominator from the exponent of the numerator.
• Apply it using the formula \( \frac{a^m}{a^n} = a^{m-n} \).

In this exercise, we apply this principle to the fraction \( \frac{10^7}{10^4} \), ensuring our result is \( 10^{3} \). This technique reduces complex expressions into simpler forms, making calculations straightforward.
It's a handy tool for students tackling exponential problems, as it transforms potential arithmetic challenges into user-friendly equations.