Problem 52 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \left(y^{-1}\right)^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\left(\frac{1}{y}\right)^{-1}\) using only positive exponents. Answer: \(\frac{1}{y}\)

Step 1: Apply the exponent rule

Notice that the given expression has a nested exponent, so we must first consider the inner exponent -1 applied to y, or \(y^{-1}\). Using the rule provided above (\(a^{-n} = \frac{1}{a^n}\)): $$ y^{-1} = \frac{1}{y^1} $$

Step 2: Apply the exponent rule once more

Now we must consider the outer -1 exponent applied to our new expression \(\frac{1}{y^1}\). Since the expression inside the parenthesis is not in the form of a simple variable raised to a negative exponent, we will apply the aforementioned rule to each of the terms in the fraction: $$ \left(\frac{1}{y^1}\right)^{-1} = \frac{1^{-1}}{(y^1)^{-1}} $$

Step 3: Simplify the expression

Finally, we can simplify the expression further using the same rules as before. For 1 raised to any power, the result is always 1, and for \(y^1\), the resulting exponent is -1 times -1, which is 1: $$ \frac{1^{-1}}{(y^1)^{-1}} = \frac{1}{y^1} $$ The final expression with only positive exponents is: $$ \frac{1}{y^1} = \frac{1}{y} $$

Key Concepts

Exponent Rules

Exponent rules are basic guidelines for handling expressions that have powers or exponents. These rules make manipulating algebraic expressions much easier. One of the primary exponent rules to remember is \(a^{-n} = \frac{1}{a^n}\). This means that any number with a negative exponent can be expressed as the reciprocal of the number raised to the opposite positive exponent.
For example, \(3^{-2}\) is calculated as \(\frac{1}{3^2} = \frac{1}{9}\). This rule applies to any number, not just integers; it works with variables as well.

• Product of Powers Rule: When you multiply like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: To raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

Understanding these rules will form a strong foundation for managing more complex expressions and algebraic problems.

Simplifying Expressions

Simplifying expressions is the process of reducing algebraic phrases for ease of use and calculation. The aim is to make them as straightforward as possible while maintaining their original value.
Here are some broad steps to simplify expressions:

• Look for and use exponent rules to create a single power or reduce powers to positive terms.
• Combine like terms, which are terms that have exactly the same variable parts. For example, \(3x\) and \(5x\) can be combined to make \(8x\).
• Eliminate any fractions by finding a common denominator or using other arithmetic techniques.

The solution applies these ideas by transforming \(y^{-1}\) into \(\frac{1}{y}\). Then, handling \((\frac{1}{y})^{-1}\) using the negative exponent rule and simplifying it back down to just \(y\). Mastery of simplification allows for trends like factorization and solving polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They serve as a bridge to advanced algebra and calculus concepts. Each part of an algebraic expression plays a specific role in the problem's overall structure.
An expression such as \(3x^2 + 4x - 5\) consists of:

• **Coefficients**: Numbers multiplying the variable part, such as 3 and 4.
• **Variables**: Usually letters that represent unknown values (x in this case).
• **Exponents**: Indicate by what power a variable is being raised (\(x^2\) means x is raised to the power of 2).
• **Constants**: Fixed numbers that do not change, like -5.

Handling algebraic expressions involves understanding how to navigate these components efficiently, using rules like distribution and collecting like terms to evaluate or simplify them. The exercise demonstrates handling nested powers within an expression, a skill important when dealing with algebraic challenges.