Problem 58 - Algebra 1

Question

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{16 x^{4}}{32 x^{8}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{16 x^{4}}{32 x^{8}}\) is \(\frac{1}{2x^{4}}\)

Step 1: Simplify the coefficients

Begin by simplifying the coefficients: 16 and 32. 16/32 simplifies to 1/2. The expression becomes \(\frac{1}{2} \cdot \frac{x^{4}}{x^{8}}\)

Step 2: Simplify the variable

Next, handle the variable with the exponents. When dividing numbers with the same base, subtract the exponents. \(x^{4}/x^{8} = x^{4-8} = x^{-4}\). The expression then becomes \(\frac{1}{2} \cdot x^{-4}\)

Step 3: Final Simplification

Finally, simplify the expression. As the exponent of x is negative, we move it to the denominator and it becomes positive. Hence \(\frac{1}{2} \cdot x^{-4}= \frac{1}{2x^{4}}\)

Key Concepts

Exponent Rules

When working with algebraic expressions, knowing exponent rules can greatly simplify the process. Exponents, or powers, are a shorthand way to express repeated multiplication. For example, \( x^3 \) means \( x \) multiplied by itself three times.

One of the fundamental rules of exponents is the 'Quotient of Powers Rule'. It states that when you divide two powers with the same base, you can subtract the exponents. In mathematical terms, \( a^m / a^n = a^{m-n} \). Applying this rule can help reduce complex expressions into simpler forms.

Another important rule is the 'Power of a Product Rule', where \( (ab)^n = a^n b^n \), which means that a product raised to an exponent is equal to each factor raised to the exponent separately.

These are just two of the various exponent rules, but they are essential for simplifying expressions efficiently.

Simplify Fractions

To simplify fractions, we seek to reduce them to their lowest terms. This requires finding the greatest common factor (GCF) of the numerator and the denominator and then dividing both by this number.

For instance, if we consider the coefficients of the fraction \( \frac{16}{32} \) from our exercise, we see that 16 is a factor of 32. Therefore, we can divide both the numerator and denominator by 16, which is the GCF in this case, to simplify the fraction to \( \frac{1}{2} \).

Simplifying fractions is not limited to numerical coefficients but also applies to algebraic terms. When algebraic terms are involved as in \( \frac{x^4}{x^8} \), we still aim to cancel common factors in both the numerator and the denominator to achieve the simplest form.

Negative Exponents

Negative exponents can cause some confusion, but they follow a straightforward rule. An expression with a negative exponent like \( x^{-n} \) is equivalent to \( 1/x^n \). In other words, you can transform a negative exponent into a positive one by moving the base to the other side of the fraction line.

Applying this to our exercise, \( x^{-4} \) can be rewritten as \( 1/x^4 \). This rule helps in simplifying expressions and is a key step to obtaining a final expression without negative exponents. Remember, the only change we make is to the position of the base; the absolute value of the exponent remains the same.

Understanding negative exponents is critical for simplifying algebraic expressions and for general algebra proficiency.