Problem 59 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive and write your answer in radical notation. $$ \sqrt[4]{x^{3}} \cdot \sqrt[3]{x} $$

Step-by-Step Solution

Answer

\( \sqrt[12]{x^{13}} \)

Step 1: Convert Radicals to Exponents

First, express each of the radicals as exponents. The expression \( \sqrt[4]{x^3} \) can be written as \( x^{3/4} \) and \( \sqrt[3]{x} \) can be represented as \( x^{1/3} \). The expression now becomes \( x^{3/4} \times x^{1/3} \).

Step 2: Use the Product Rule for Exponents

Apply the product rule for exponents, which is \( a^m \times a^n = a^{m+n} \). We need to add the exponents: \( x^{3/4} \) + \( x^{1/3} \).

Step 3: Find a Common Denominator for the Exponents

The exponents have different denominators, so find a common denominator. The least common multiple of 4 and 3 is 12. Convert \( \frac{3}{4} \) to \( \frac{9}{12} \) and \( \frac{1}{3} \) to \( \frac{4}{12} \).

Step 4: Add the Fractions

Now, add the fractions: \( \frac{9}{12} + \frac{4}{12} = \frac{13}{12} \). The expression becomes \( x^{13/12} \).

Step 5: Convert the Exponent Back to Radical Notation

Now, write the exponent \( \frac{13}{12} \) back into radical notation. The expression \( x^{13/12} \) can be written as \( \sqrt[12]{x^{13}} \). Therefore, the simplified expression is \( \sqrt[12]{x^{13}} \).

Key Concepts

Radical Notation

Radical notation is a way to represent roots, such as square roots, cube roots, and others. Instead of writing roots in fractional exponent form, we use radical symbols. For example, the square root of a number \( x \) is written as \( \sqrt{x} \), while the cube root is written as \( \sqrt[3]{x} \).
In our exercise, we started with two expressions: \( \sqrt[4]{x^3} \) and \( \sqrt[3]{x} \). These are written in radical notation using the root symbols with numbers to indicate which root to take.

• The index of the root is the small number outside and to the left of the radical sign.
• The radicand is the number or expression inside the radical sign.

When simplifying expressions, it's common to convert from radical notation to exponent form, because exponent rules are simpler to apply than dealing with complex roots.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. An exponent says how many times you use a number in a multiplication. For example, \( x^3 \) means \( x \times x \times x \).
When dealing with radicals, these can be expressed in terms of exponents, often called fractional exponents. For instance, the fourth root \( \sqrt[4]{x^3} \) can be written as \( x^{3/4} \).

• Numerator of the fraction (3 in \( \frac{3}{4} \)) tells how many times the radicand is multiplied.
• Denominator (4) tells which root is being taken.

Converting radicals to exponents allows us to apply exponent rules, which can simplify calculations and help solve more complex expressions.

Product Rule

The product rule is an exponent rule that simplifies expressions where the same base number is raised to different powers. The product rule states: \( a^m \times a^n = a^{m+n} \). This means, when you multiply like bases, you add the exponents together.
In our exercise, after converting to exponent form, we got \( x^{3/4} \times x^{1/3} \). To apply the product rule, we first needed a common denominator to add the fractions \( \frac{3}{4} \) and \( \frac{1}{3} \).

• Find a common denominator, in this case, 12.
• Convert \( \frac{3}{4} \) to \( \frac{9}{12} \) and \( \frac{1}{3} \) to \( \frac{4}{12} \).
• Add these fractions to get \( \frac{13}{12} \).

Thus, applying the product rule helps us simplify the expression to a single exponent term \( x^{13/12} \), which we can then convert back to radical notation.