Problem 66 - Elementary Algebra

Question

\(\frac{\left(6 \times 10^{3} \mathrm{in} .\right)^{2}}{6 \times 10^{4} \mathrm{in} .}\)

Step-by-Step Solution

Answer

600 in

Step 1: Simplify the Numerator

First, simplify the numerator \[\frac{(6 \times 10^3 \text{ in})^2}{6 \times 10^4 \text{ in}} \]. Calculate \[((6 \times 10^3 \text{ in}))^2 \]. This gives \[6^2 \times (10^3)^2 \text{ in}^2 = 36 \times 10^{6} \text{ in}^2 \].

Step 2: Rewrite the Expression

Rewrite the expression using the simplified numerator: \[ \frac{36 \times 10^6 \text{ in}^2}{6 \times 10^4 \text{ in}} \]

Step 3: Cancel Common Terms

Cancel the common factor of 6 from the numerator and denominator: \[ \frac{36 \times 10^6 \text{ in}^2}{6 \times 10^4 \text{ in}} = \frac{36}{6} \times \frac{10^6 \text{ in}^2}{10^4 \text{ in}} = 6 \times \frac{10^6 \text{ in}^2}{10^4 \text{ in}} \]

Step 4: Simplify the Exponent

Simplify the exponent by subtracting 4 from 6: \[ 6 \times \frac{10^6 \text{ in}^2}{10^4 \text{ in}} = 6 \times 10^{6-4} \text{ in} = 6 \times 10^2 \text{ in} \]

Step 5: Final Answer

The simplified expression is: \[ 6 \times 10^2 \text{ in} = 600 \text{ in} \]

Key Concepts

Exponent rules

Exponent rules are key in solving many algebraic problems. Understanding these rules helps simplify complex expressions with powers. Here are some basic rules:

• Product of Powers: \[ a^m \times a^n = a^{m+n} \] When multiplying like bases, add the exponents.
• Power of a Power: \[ (a^m)^n = a^{m \times n} \] When raising a power to another power, multiply the exponents.
• Quotient of Powers: \[ \frac{a^m}{a^n} = a^{m-n} \] When dividing like bases, subtract the exponents.

For example, in the given exercise, we use the power of a power rule to simplify the numerator and the quotient of powers rule to further simplify the expression.

Simplifying expressions

Simplifying expressions means reducing them to their simplest form. Follow these steps:

• Combine like terms (terms with the same variable and exponent).
• Apply exponent rules to manage terms with powers.
• Factor where possible to cancel common factors.

In the provided exercise, we started by simplifying the numerator: \[ (6 \times 10^3 \text{ in})^2 \] This involves squaring both the coefficient and the term with an exponent: \[ 6^2 \times 10^{3 \times 2} = 36 \times 10^6 \text{ in}^2 \] By simplifying the expression step by step, each part of the expression becomes more manageable. This approach ensures accuracy and clarity.

Algebraic fractions

Working with algebraic fractions involves simplifying them while adhering to algebraic rules. An algebraic fraction has a numerator and a denominator, both of which may contain variables and exponents. Follow these principles:

• Factor to Simplify: Factor out common terms to cancel them, reducing the complexity of the fraction.
• Cancel Common Factors: Cancel factors in the numerator and denominator to simplify the fraction.

In the exercise, we simplify \[ \frac{36 \times 10^6 \text{ in}^2}{6 \times 10^4 \text{ in}} \] by cancelling the common factor of 6: \[ \frac{36}{6} \times \frac{10^6}{10^4} = 6 \times \frac{10^6}{10^4} \] This step reduces the fraction to a simpler form, making further simplification easier.

Scientific notation

Scientific notation is a way to express very large or very small numbers conveniently using powers of ten. This is especially useful in algebra for simplifying calculations. The general form is: \[ a \times 10^n \] where \( 1 \leq \lvert a \rvert \< 10 \) and \( n \) is an integer. Follow these guidelines:

• Multiplying in Scientific Notation: Multiply the base numbers and add the exponents.
• Dividing in Scientific Notation: Divide the base numbers and subtract the exponents.

In the exercise, the final expression \[ 6 \times 10^2 \] is a simplified form using scientific notation, making it easier to interpret the result as \ 600 \ in.