Problem 93 - College Algebra Essentials

Question

In Exercises \(91-100,\) simplify using properties of exponents. $$\frac{20 x^{\frac{1}{2}}}{5 x^{\frac{1}{4}}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(4 x^{\frac{1}{4}}\).

Step 1: Simplifying Numerical Coefficients

Divide numerical coefficients \(20\) and \(5\) to get \(4\). So, we have \(4 x^{\frac{1}{2}}/ x^{\frac{1}{4}}\).

Step 2: Applying the Rule of Exponents

Given the identical bases, use the properties of exponents to subtract the exponents: \(\frac{1}{2} - \frac{1}{4} = \frac{1}{4}\). Hence \(4 x^{\frac{1}{4}}\).

Step 3: Write the Final Answer

After simplifying, the final answer is \(4 x^{\frac{1}{4}}\)

Key Concepts

Simplifying Expressions

We often encounter complex algebraic expressions in mathematics, which can be challenging to interpret. To make it easier, we use the process of simplification, which involves rewriting these expressions in a more manageable form without changing their value. Simplification can include combining like terms, reducing fractions, or using properties of exponents and radicals.

For instance, consider the expression \frac{20 x^{\frac{1}{2}}}{5 x^{\frac{1}{4}}}\. This appears complicated at first, but by simplifying the numerical coefficients and applying the rules of exponents, which is breaking down the expression into smaller and more interpretable parts, we can easily understand and solve it. Thus simplification turns a daunting expression into a concise and more understandable one like \(4 x^{\frac{1}{4}}\).

Exponent Rules

When dealing with exponents, it is crucial to understand and apply a set of rules that govern their behavior. These rules help us simplify expressions with exponential terms efficiently. Some primary exponent rules include the product rule \(a^m \cdot a^n = a^{m+n}\), the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\), the power rule \((a^m)^n = a^{m\cdot n}\), and the rule for zero exponent \(a^0 = 1\), where \(a\) is a nonzero base and \(m\) and \(n\) are exponents.

Using the quotient rule in our example, we subtract the exponents of like bases: \(\frac{1}{2} - \frac{1}{4} = \frac{1}{4}\). This straightforward use of the exponent rules often leads to a significant simplification of algebraic expressions.

Radical Expressions

Radical expressions involve roots of numbers or expressions, such as square roots, cube roots, etc. A radical expression like \(\sqrt{x}\) can also be written as an exponent, \(x^{\frac{1}{2}}\). Understanding the connection between radicals and exponents is vital for simplifying radical expressions, which often involves rationalizing the denominator or consolidating like terms.

In the context of our example, seeing \(x^{\frac{1}{2}}\) and \(x^{\frac{1}{4}}\) as square root and fourth root of \(x\) respectively, can provide insight into how these radicals relate and combine according to the rules of exponents. Therefore, mastering radical expressions allows you to navigate seamlessly between different forms of the same mathematical concept.

Algebraic Properties

Algebra relies on a set of properties that allows for the consistent manipulation of expressions and equations. These properties include commutative, associative, distributive, as well as properties of equality and inequality. Knowing these properties enables one to rearrange and combine terms in a variety of ways to solve algebraic problems efficiently.

Applying these algebraic properties in our given problem, we use the associative property to group and divide the coefficients separately from the variables with exponents. We also implicitly use the distributive property when dealing with exponents across multiplication or division. These properties ensure we obtain a valid and simplified version of the original complex expression, ultimately arriving at \(4 x^{\frac{1}{4}}\).