Problem 37 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \frac{x^{9}(2 x)^{4}}{x^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(16x^{10}\).

Step 1: Distribute the Exponent

Start by distributing the exponent 4 in the expression \((2x)^4\). This means you apply the exponent to both 2 and \(x\). Thus, \((2x)^4 = 2^4 \cdot x^4\). Calculate: \(2^4 = 16\) and \(x^4 = x^4\). Therefore, \((2x)^4 = 16x^4\).

Step 2: Substitute Distributed Values into Expression

Substitute the distributed values back into the original expression. This gives you \(\frac{x^9 \cdot 16x^4}{x^3}\).

Step 3: Simplify the Numerator

In the numerator, combine the exponents of \(x^9\) and \(x^4\). When multiplying with the same base, add the exponents: \(x^9 \cdot x^4 = x^{9+4} = x^{13}\). The numerator simplifies to \(16x^{13}\).

Step 4: Divide by the Denominator

Now, divide the expression \(16x^{13}\) by \(x^3\). Subtract the exponents: \(x^{13-3} = x^{10}\). Thus, the expression simplifies to \(16x^{10}\).

Step 5: Write the Final Answer

Since there are no negative exponents in the expression \(16x^{10}\), this is your final simplified expression.

Key Concepts

Negative Exponents

Negative exponents can often seem tricky, but they are actually pretty straightforward once you understand what they mean. Negative exponents indicate the reciprocal of a number. For example, if you have the expression \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). This means you're taking the base \(a\) and flipping it to the denominator with a positive exponent.In terms of simplifying expressions, it’s usually best to eliminate negative exponents. Why? Because negative exponents can make expressions look complicated, and it's generally more intuitive working with positive exponents. Always remember:

• A negative exponent flips the base to the opposite side of a fraction.
• If the base is in the numerator, move it to the denominator, and vice versa.
• Apply the absolute value of the exponent as the new power.

By following these simple rules, you'll find it much easier to handle algebraic fractions and simplify them cleanly.

Distributing Exponents

Distributing exponents involves applying an exponent to each factor inside a parenthesis. This concept is essential when you encounter expressions like \((ab)^n\). Here, the exponent \(n\) applies to both \(a\) and \(b\), so this expression becomes \(a^n \cdot b^n\).In the context of the provided exercise, we distributed the exponent of 4 over the term \((2x)^4\). This means:

• The number 2 is raised to the power of 4, giving \(2^4 = 16\).
• The variable \(x\) is raised to the power of 4, resulting in \(x^4\).

Therefore, \((2x)^4\) becomes \(16x^4\). This step is crucial because it simplifies the process of combining and reducing exponents in subsequent steps. Always ensure every factor within the parenthesis receives the same exponent.

Combining Exponents

When you have expressions that involve the same base being multiplied, you can easily simplify them by combining the exponents. This is because of the exponent rule: when you multiply two powers with the same base, you add the exponents.Let's break it down with the exercise example:\[x^9 \cdot x^4 = x^{9+4} = x^{13}\]This tells us that you simply add the exponents 9 and 4 to get 13. So, \(x^9\), when multiplied by \(x^4\), becomes \(x^{13}\).Now, if you have a similar situation while dividing, you would subtract the exponents instead:

• Multiplying: Add exponents \(x^m \cdot x^n = x^{m+n}\).
• Dividing: Subtract exponents \(\frac{x^m}{x^n} = x^{m-n}\).

By consistently applying these rules, you can simplify complex algebraic expressions accurately and efficiently.

Algebraic Fractions

Algebraic fractions operate much like numerical fractions, but with variables. Simplifying them often involves reducing terms, distributing, eliminating negative exponents, and combining similar bases.From the exercise, after distributing the exponent, we end up with \(\frac{x^9 \cdot 16x^4}{x^3}\). Now, focus on simplifying this fraction:\[\frac{x^9 \cdot 16x^4}{x^3} = \frac{16x^{13}}{x^3}\]Here, you apply the rule of subtracting exponents in the denominator from those in the numerator, resulting in \(x^{13-3} = x^{10}\).Key tips for working with algebraic fractions:

• Look out for common factors in the numerator and denominator that can be cancelled out.
• Apply exponent rules (combining and distributing) to simplify expressions.
• Maintain a positive exponent for a cleaner and more interpretable expression.

These steps enable you to transform complex algebraic fractions into their simplest form, making them much easier to work with and understand.