Problem 46 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\frac{k^{-3 / 5} h^{-1 / 3} t^{2 / 5}}{k^{-1 / 5} h^{-2 / 3} t^{1 / 5}}$$

Step-by-Step Solution

Answer

\(\frac{h^{1/3} t^{1/5}}{k^{2/5}}\)

Step 1: Apply the Quotient Rule for Exponents

The quotient rule states that when you divide like bases, you subtract the exponents: \(a^m / a^n = a^{m-n}\). Apply this rule separately to each variable: \(k\), \(h\), and \(t\).

Step 2: Simplify the Exponent for k

Subtract the exponents for \(k\): \(-\frac{3}{5} - (-\frac{1}{5}) = -\frac{3}{5} + \frac{1}{5} = -\frac{2}{5}\).

Step 3: Simplify the Exponent for h

Subtract the exponents for \(h\): \(-\frac{1}{3} - (-\frac{2}{3}) = -\frac{1}{3} + \frac{2}{3} = \frac{1}{3}\).

Step 4: Simplify the Exponent for t

Subtract the exponents for \(t\): \(\frac{2}{5} - \frac{1}{5} = \frac{1}{5}\).

Step 5: Write with Positive Exponents

The expression now is \(k^{-2/5} h^{1/3} t^{1/5}\). To write the expression with positive exponents only, move \(k^{-2/5}\) to the denominator: \(\frac{h^{1/3} t^{1/5}}{k^{2/5}}\).

Step 6: Final Answer

The final expression with positive exponents is \(\frac{h^{1/3} t^{1/5}}{k^{2/5}}\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. They consist of a base and an exponent. The base is the number being multiplied, and the exponent indicates how many times it is multiplied by itself.
For example, in the expression \(a^b\), \(a\) is the base, and \(b\) is the exponent. This means you multiply \(a\) by itself \(b\) times: \(a \times a \times a \times \ldots\) (\(b\) times).
When dealing with exponents in algebraic expressions, it's important to know a few basic rules:

• Multiplication: \(a^m \times a^n = a^{m+n}\)
• Division (Quotient Rule): \(a^m / a^n = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \times n}\)
• Negative exponent: \(a^{-n} = 1/a^n\)

These rules help simplify expressions and make calculations more manageable.

Quotient Rule

The quotient rule of exponents states that when you divide two exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator. Essentially, it simplifies the process of dividing powers.
For example, in the expression \(\frac{a^m}{a^n}\), the quotient rule allows you to write it as \(a^{m-n}\).
This rule is useful for simplifying expressions with exponents, especially when working with variables. In the given exercise, this rule is applied separately to the variables \(k\), \(h\), and \(t\):

• For \(k\), we subtract: \(-\frac{3}{5} - (-\frac{1}{5})\)
• For \(h\), we subtract: \(-\frac{1}{3} - (-\frac{2}{3})\)
• For \(t\), we subtract: \(\frac{2}{5} - \frac{1}{5}\)

This method systematically reduces the exponents and allows us to work with simpler expressions.

Simplification of Expressions

Simplification of expressions involves reducing an expression to its simplest form. In the context of algebraic expressions with exponents, simplification often means:

• Combining like terms
• Applying exponent rules (such as the quotient rule)
• Rewriting expressions to have only positive exponents

The original expression \(\frac{k^{-3/5} h^{-1/3} t^{2/5}}{k^{-1/5} h^{-2/3} t^{1/5}}\) was simplified by applying the quotient rule to each variable and converting negative exponents to positive.
Negative exponents can make expressions appear more complex. Using the rule \(a^{-n} = 1/a^n\), we can rewrite them as positive by moving the term to the denominator. This gives us a final expression of \(\frac{h^{1/3} t^{1/5}}{k^{2/5}}\), which is more easily understandable and visually cleaner, as it uses positive exponents only. Simplification not only helps in reducing computational complexity but also enhances clarity when representing mathematical ideas.