Problem 79 - Algebra 2 and Trigonometry

Question

In \(74-82,\) write each expression as a power with positive exponents in simplest form. $$ \frac{\sqrt[6]{a^{5}}}{\sqrt[5]{a^{5}}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{\sqrt[6]{a}}\)."

Step 1: Rewrite Using Rational Exponents

The root of a number can be expressed using rational exponents. In particular, \(\sqrt[n]{x^m} = x^{m/n}\). Thus, rewrite the expression \(\frac{\sqrt[6]{a^{5}}}{\sqrt[5]{a^{5}}}\) as \(\frac{a^{5/6}}{a^{5/5}}\).

Step 2: Simplify Exponents in the Denominator

Calculate the exponent in the denominator: \(a^{5/5} = a^1\). This simplifies the denominator to \(a\). Thus, the expression becomes \(\frac{a^{5/6}}{a}\).

Step 3: Subtract the Exponents

Subtract the exponent in the denominator from the exponent in the numerator using the rule \(\frac{x^m}{x^n} = x^{m-n}\). Subtracting the exponents gives \(a^{5/6 - 1}\).

Step 4: Simplify the Exponent

Perform the subtraction: \(5/6 - 1 = 5/6 - 6/6 = -1/6\). Hence, the expression simplifies to \(a^{-1/6}\).

Step 5: Write with Positive Exponents

Rewrite \(a^{-1/6}\) with a positive exponent. This gives \(\frac{1}{a^{1/6}}\).

Step 6: Write as a Root

Since \(a^{1/6}\) can be written as \(\sqrt[6]{a}\), the expression in simplest form is \(\frac{1}{\sqrt[6]{a}}\).

Key Concepts

Rational Expressions

Rational expressions are mathematical expressions that resemble fractions, but instead of integers, they involve polynomials or variables with exponents. A rational expression takes the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). In a simpler sense, it is an expression where both the numerator and the denominator contain variables.Understanding rational expressions is crucial because they help in operations such as addition, subtraction, multiplication, and division of algebraic expressions. They also play a significant role when dealing with complex equations by allowing the simplification process between different terms more manageable.
One important aspect of working with rational expressions is ensuring the denominator never equals zero. This is because division by zero is undefined in mathematics. Therefore, part of simplifying these expressions involves locating and stating any restrictions on the variable's value.

Exponent Laws

Exponent laws describe the rules that make operations with powers consistent and predictable. These rules are essential when it comes to simplifying expressions that involve powers. There are several key exponent laws:- **Product of Powers:** When multiplying terms with the same base, add their exponents. For example, \( x^m \times x^n = x^{m+n} \).- **Quotient of Powers:** When dividing terms with the same base, subtract the exponents. When you simplify a fraction like \( \frac{x^m}{x^n} = x^{m-n} \), this rule is crucial.
- **Power of a Power:** When raising a power to another power, multiply the exponents, like \( (x^m)^n = x^{m \times n} \).
- **Zero Exponent:** Any non-zero base raised to zero equals one, \( x^0 = 1 \).
- **Negative Exponent:** A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent, \( x^{-n} = \frac{1}{x^n} \).These laws help in converting roots into rational exponents, which is particularly useful when simplifying expressions with roots or when writing expressions with positive exponents.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form without changing their value. This process often involves several mathematical techniques, including combining like terms, using distributive properties, and applying exponent rules. When dealing with expressions having exponents, simplifying can be highly beneficial in understanding, solving, and representing equations efficiently.First, you may need to convert any roots into rational exponents. For example, \( \sqrt[n]{x^m} \) can be expressed as \( x^{m/n} \). This transformation eases manipulation since exponent laws can now be applied to simplify further.After expressing everything in terms of rational exponents, use the quotient of powers law to simplify the fraction. This means subtracting the exponents if the bases are the same. If any negative exponents appear, convert them back to positive by taking the reciprocal.Ultimately, simplifying expressions not only helps in solving equations but also makes it easier to identify domains and extract crucial information from mathematical models.