Problem 576 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}}\) (b) \(\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}} \odot \frac{m^{\frac{13}{8}}}{m^{\frac{5}{8}}}\)

Step-by-Step Solution

Answer

(a) t (b) xm

Step 1: Simplify the first expression

To simplify \(\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}}\), use the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \). Subtract the exponents: \(\frac{12}{5} - \frac{7}{5} = \frac{5}{5} = 1\). So, \(\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}} = t^1 = t\).

Step 2: Simplify the first part of the second expression

For \(\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}}\), again use the property of exponents. Subtract the exponents: \(\frac{3}{2} - \frac{1}{2} = \frac{2}{2} = 1\). So, \(\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}} = x^1 = x\).

Step 3: Simplify the second part of the second expression

Next, simplify \(\frac{m^{\frac{13}{8}}}{m^{\frac{5}{8}}}\) in a similar fashion. Subtract the exponents: \(\frac{13}{8} - \frac{5}{8} = \frac{8}{8} = 1\). So, \(\frac{m^{\frac{13}{8}}}{m^{\frac{5}{8}}} = m^1 = m\).

Step 4: Combine the results

From the results of Step 2 and Step 3, multiply the simplified parts together: \(x \times m = xm\).

Key Concepts

Properties of Exponents

When working with exponents, certain properties make it easier to manipulate and simplify expressions. One fundamental property is the quotient rule, which states that for any nonzero number \(a\) and integers \(m\) and \(n\), \(\frac{a^m}{a^n} = a^{m-n}\). This property helps us remove fractions in the exponents by turning them into simpler expressions.
Here are a few key properties of exponents to remember:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^m = a^m \cdot b^m\)
• Power of a Quotient: \((\frac{a}{b})^m = \frac{a^m}{b^m}\)

In our exercise, we applied the quotient rule to simplify expressions by subtracting the exponents step by step.

Fractional Exponents

Fractional exponents represent both an exponent and a root. For example, \(a^{\frac{m}{n}}\) can be written as \(\sqrt[n]{a^m}\). This means that any fractional exponent is the result of raising the base to a power and then taking a root.
Here's how to think about fractional exponents:

• \(a^{\frac{1}{2}}\) is the square root of \(a\).
• \(a^{\frac{1}{3}}\) is the cube root of \(a\).
• \(a^{\frac{m}{n}}\) means first raising \(a\) to the \(m\)-th power and then taking the \(n\)-th root.

In the exercise provided, we subtract fractions when simplifying expressions. To simplify \(\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}}\), we notice that both exponents have the same denominator, making subtraction straightforward: \(\frac{12}{5} - \frac{7}{5} = 1\). This logical step helps simplify seemingly complex fractions.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form using various rules and properties of mathematics. This often includes combining like terms, factoring, and canceling out common factors.
In our solved exercise, simplification was carried out in multiple steps:

• First, simplify \(\frac{t^{12/5}}{t^{7/5}}\) using the properties of exponents.
• Then, individually simplify each fraction in the second expression \(\frac{x^{3/2}}{x^{1/2}} \odot \frac{m^{13/8}}{m^{5/8}}\).
• Finally, combine the results to get the final simplified form \(xm\).

Each of these steps involves reducing fractions by using the exponent subtraction rule and multiplying the results. By breaking down complex expressions into smaller, manageable steps, we make algebraic simplification coherent and less intimidating for students.