Question

Decide which expressions are equiva lent. Assume all variables are positive. (a) \(\left(\frac{2}{3}\right)^{-n}\) (b) \(\left(\frac{1}{\frac{2}{3}}\right)^{n}\) (c) \(\left(\frac{3}{2}\right)^{n}\) (d) \(-\left(\frac{2}{3}\right)^{n}\) (e) \(\frac{2^{-n}}{3^{-n}}\)

Short answer

Based on the simplification and comparison of the given expressions, we found out that expressions (b), (c), and (e) are equivalent to each other. On the other hand, expressions (a) and (d) are not equivalent to the others or each other.

Step-by-step solution

Simplify the expressions

Let's simplify each expression: (a) \(\left(\frac{2}{3}\right)^{-n}\) We know that \(a^{-n}=\frac{1}{a^n}\). So, we get: \(\left(\frac{2}{3}\right)^{-n}= \frac{1}{\left(\frac{2}{3}\right)^{n}}\) (b) \(\left(\frac{1}{\frac{2}{3}}\right)^{n}\) We can simplify the base by multiplying both the numerator and denominator by 3, which results in: \(\left(\frac{3}{2}\right)^{n}\) (c) \(\left(\frac{3}{2}\right)^{n}\) This expression does not require simplification. (d) \(-\left(\frac{2}{3}\right)^{n}\) Since there is a negative sign in the expression, this cannot be equal to any other expressions. (e) \(\frac{2^{-n}}{3^{-n}}\) We can rewrite this by using the property \(a^{-n}=\frac{1}{a^n}\), resulting in: \(\frac{\frac{1}{2^n}}{\frac{1}{3^n}}\) Now, we can invert and multiply: \(\frac{1}{2^n} \cdot \frac{3^n}{1} = \frac{3^n}{2^n} = \left(\frac{3}{2}\right)^n\)

Compare the simplified expressions

Now we can compare the expressions to determine which are equivalent: - (a) \(\frac{1}{\left(\frac{2}{3}\right)^{n}}\) - (b) \(\left(\frac{3}{2}\right)^{n}\) - (c) \(\left(\frac{3}{2}\right)^{n}\) - (d) \(-\left(\frac{2}{3}\right)^{n}\) (Cannot be equivalent to others due to the negative sign) - (e) \(\left(\frac{3}{2}\right)^n\) We can see that after simplifying the expressions, (b), (c), and (e) are all equal to \(\left(\frac{3}{2}\right)^{n}\) and therefore are equivalent to each other. The expression (a) and (d) are not equivalent to each other or the other expressions.

Key concepts

Equivalent Expressions

Equivalent expressions are different algebraic phrases that denote the same value for every substitution of the variable(s). This concept is essential because it helps solve equations and simplify expressions for better understanding. When we talk about equivalent expressions, we look for expressions that result in the same value after simplification or evaluation.

For instance, in the exercise above, expressions (b), (c), and (e) were determined to be equivalent after simplification. Each of these expressions simplifies to \( \left(\frac{3}{2}\right)^n \), meaning they'd yield the same result for any positive value of \( n \). Comparing expressions in this way involves using algebraic properties to identify patterns and similarities. Identifying equivalent expressions allows us to simplify problems and choose the most convenient form for calculations.

Negative Exponents

Negative exponents are an interesting aspect of algebra that can initially seem complex, but they're useful once understood. A negative exponent indicates reciprocal action in an expression. Specifically, \( a^{-n} \) means \( \frac{1}{a^n} \). Essentially, you flip the base over and change the sign of the exponent to positive.

In the given exercise, expression (a) \( \left(\frac{2}{3}\right)^{-n} \) uses a negative exponent. By applying the rule of reciprocal action, we simplify this to \( \frac{1}{\left(\frac{2}{3}\right)^{n}} \). This process allows us to transform and better understand expressions, ensuring that no matter the form or complexity, they can be simplified to a standard form. Negative exponents are essentially a way of expressing values smaller than one in fraction form, which is a handy mathematical tool.

Rational Expressions

Rational expressions are a key part of algebra consisting of a ratio of two polynomials. In other words, it's like a fraction but with polynomials in the numerator and/or denominator. Understanding how to manipulate rational expressions is essential for many algebraic processes, such as simplifying expressions, performing operations, and solving equations.

In our exercise, expressions such as (e) \( \frac{2^{-n}}{3^{-n}} \) involve rational expressions with polynomials raised to negative powers. Simplifying such expressions often involves converting these negative exponents, as we simplified \( \frac{2^{-n}}{3^{-n}} \) to \( \left(\frac{3}{2}\right)^n \) using reciprocal and multiplication rules. By understanding and mastering the properties of rational expressions, including those with fractional and negative powers, one can solve a wide range of algebraic problems with ease.