Problem 48 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(\frac{2 s^{3}}{r t^{4}}\right)^{-5}$$

Step-by-Step Solution

Answer

\[\frac{r^5 t^{20}}{32s^{15}}\]

Step 1: Apply Negative Exponent Property

Negative exponents can be simplified by reciprocating the base and taking its positive power. Using this rule, we can rewrite our expression as follows: \[\left(\frac{2 s^{3}}{r t^{4}}\right)^{-5} = \left(\frac{r t^4}{2 s^3}\right)^5\]

Step 2: Apply Power Rules

Apply the power rules to distribute the power of 5 to each individual element within the expression. The power rules state that \((ab)^n=a^n b^n\), and \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), for all positive integers n. \[\left(\frac{r t^4}{2 s^3}\right)^5 = \frac{r^5 t^{4\cdot5}}{(2^5 s^{3\cdot5})}\]

Step 3: Simplify

Finally, simplify the expression by doing the exponent calculations: \[\frac{r^5 t^{20}}{(2^5 s^{15})} = \frac{r^5 t^{20}}{32s^{15}}\] And there you have it! The simplified expression without any negative exponents is: \[\frac{r^5 t^{20}}{32s^{15}}\]

Key Concepts

Negative Exponents

Negative exponents may seem confusing, but they are actually quite straightforward. When faced with a negative exponent, imagine it as a signal to "flip" the base and make the exponent positive. For instance, writing \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\). This means that the negative exponent tells you to take the reciprocal of the base and then apply the positive exponent.
Let's look at an example. Suppose you have \((\frac{2}{3})^{-2}\). According to the negative exponent rule, you would first flip the fraction: \((\frac{3}{2})^{2}\). Now you can apply the positive exponent, leading to the final result of \(\frac{9}{4}\).

• Remember: The negative exponent inverts the number to which it is applied.
• Convert negative exponents by flipping the base, making the exponent positive.

Power Rules in Algebra

Power rules in algebra are essential for simplifying expressions, especially when dealing with exponents. These rules help determine how exponents are distributed over variables and numbers. One fundamental rule states that \((ab)^n = a^n \cdot b^n\), meaning that when you have a product raised to an exponent, you can distribute the exponent to each factor.
Another essential rule is the division rule for exponents: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This means if you have a fraction raised to a power, you can apply the exponent to both the numerator and denominator.
To illustrate, consider \(\left(\frac{3x}{4y^2}\right)^3\). Applying the power rules, you get \(\frac{3^3 \cdot x^3}{4^3 \cdot (y^2)^3}\), which simplifies to \(\frac{27x^3}{64y^6}\).

• Power rules allow you to handle exponents systematically.
• They are key for simplifying algebraic expressions.

Nonzero Real Numbers

In algebra, nonzero real numbers are crucial because they ensure valid operations and avoid undefined conditions. A real number is any value found on the number line, which includes both rational and irrational numbers. When we say a number is "nonzero," it simply means it is not equal to zero. This is important because dividing by zero is undefined in mathematics.
For example, if you have the expression \(\frac{1}{x}\), \(x\) must be a nonzero real number to ensure the expression remains valid. In algebraic expressions, variables often represent nonzero real numbers to avoid any issues related to undefined terms.

• Nonzero real numbers are any numbers that are not zero.
• They ensure the validity of algebraic operations.