Question

In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(\frac{1}{e}\right)^{-2}\) (b) \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\) (c) \(\frac{e^{5}}{e^{3}}\) (d) \(\frac{1}{e^{-3}}\)

Short answer

(a) \(e^2\), (b) \(\frac{1}{e^3}\), (c) \(e^2\), (d) \(e^3\)

Step-by-step solution

Simplify the expression using property of negative exponents

For (a) \(\left(\frac{1}{e}\right)^{-2}\), use the property of negative exponent which states that \(a^{-n} = \frac{1}{a^n}\). Following this principle, the expression becomes \(e^2\).

Apply the rule to simplify the expression where additional operation of division is involved

For (b) \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\), use the property of division which states that \(\frac{a^{m}}{a^n} = a^{m-n}\). Then, you apply the rule of negative exponents. The expression becomes \(e^{2-5} = e^{-3}\), and then it turns into \(\frac{1}{e^3}\) by applying the rule of negative exponents.

Apply the subtracting exponents rule to simplify the expression

For (c) \(\frac{e^{5}}{e^{3}}\), use the property of division which states that \(\frac{a^{m}}{a^n} = a^{m-n}\). The expression simplifies to \(e^{5-3} = e^2\).

Simplify the expression using property of negative exponents

For (d) \(\frac{1}{e^{-3}}\) again use the property of negative exponent. It simplifies to \(e^3\).

Key concepts

Negative Exponents

Understanding negative exponents is essential for algebraic manipulation. Essentially, a negative exponent indicates that the base should be reciprocated and then raised to the absolute value of the exponent. For instance, if you have an expression like \( a^{-n} \), this would translate to \( \frac{1}{a^n} \). This rule helps to simplify expressions by turning divisions into multiplications by reciprocals, making the operation easier to handle.

Let's apply this to the given exercise. Take the expression \( \left(\frac{1}{e}\right)^{-2} \). According to the negative exponent rule, it simplifies to \( e^2 \). It's the reciprocal of \( \frac{1}{e} \) raised to the power of 2. This transformation from a negative to a positive exponent not only makes the expression simpler but also readies it for further algebraic steps if needed.

Exponent Rules

The rules of exponents are the guiding principles that allow us to manipulate expressions with exponents efficiently. These include multiplying and dividing powers with the same base, and raising a power to another power. Notably, when you divide powers with the same base, you subtract the exponents. For example, \( \frac{a^m}{a^n} = a^{m-n} \). Similarly, when you raise one power to another, like \( (a^m)^n \), you multiply the exponents to get \( a^{m \cdot n} \).

In our problem set, for the exercise regarding \( \left(\frac{e^{5}}{e^{2}}\right)^{-1} \), we first use the rule of division to get \( e^{5-2} \), which simplifies to \( e^3 \). Then we apply the negative exponent rule and end up with \( \frac{1}{e^3} \). These rules are the bread and butter of dealing with exponents in algebra.

Simplifying Expressions

The goal in simplifying expressions is to make them as straightforward as possible, often using the least number of terms to express the same value. This can involve combining like terms, factoring, expanding or using exponent rules. Simplification can make understanding, using, and communicating mathematical expressions far more manageable.

As an example from the exercise solutions, by applying the rules of exponents to \( \frac{e^{5}}{e^{3}} \), we simplify the expression to \( e^{2} \), boiling down the division to a single term with a reduced exponent. Simplification gives us the clearest picture of what the expression represents and primes it for any further operations or evaluations.

Algebraic Manipulation

Algebraic manipulation encompasses various techniques used to transform mathematical expressions into different, often simpler forms. It is a crucial skill for solving equations and understanding algebraic relationships. This includes distributing, combining like terms, using exponent rules, and factoring among others. Mastery of algebraic manipulation allows for a more profound understanding and solving of complex problems with ease.

To demonstrate this with our exercises, in the case of \( \frac{1}{e^{-3}} \), we manipulate the expression by negating the exponent to flip the fraction, simplifying to \( e^3 \). Such manipulations are a cornerstone of algebra and are used extensively in more advanced mathematical topics.