Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\frac{15 \sqrt[4]{125}}{\sqrt[4]{5}}\)

Step-by-Step Solution

Answer

The simplified expression is \(15 \sqrt{5}\).

Step 1 - Simplify the Radicals

First, simplify the radicals in the expression. Note that \( \sqrt[4]{125} = \sqrt[4]{5^3} = 5^{3/4} \)and \( \sqrt[4]{5} = 5^{1/4} \).

Step 2 - Substitute the Simplified Radicals into the Expression

Replace the radical expressions with their simplified forms in the fraction, so we have: \(\frac{15 \times 5^{3/4}}{5^{1/4}}.\)

Step 3 - Apply the Law of Exponents

Use the law of exponents which states that \(\frac{a^m}{a^n} = a^{m-n}\). Apply this to the expression:\(\frac{15 \times 5^{3/4}}{5^{1/4}} = 15 \times 5^{3/4 - 1/4} = 15 \times 5^{2/4} = 15 \times 5^{1/2}.\)

Step 4 - Simplify Further

Now simplify the exponent \(5^{1/2}\), which is equal to \(\sqrt{5}\). Therefore, the original expression simplifies to:\(15 \times \sqrt{5}.\)

Key Concepts

Understanding Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, or in this case, fourth roots. They can often look daunting, but the key to simplifying them is to break them down into their base and exponent. In the expression \( \sqrt[4]{125} \), note that 125 can be written as \( 5^3 \). Hence, \( \sqrt[4]{125} \) turns into \( \sqrt[4]{5^3} \), which is equal to \( 5^{3/4} \).
Radicals like \( \sqrt[4]{5} \) can also be transformed using exponents, becoming \( 5^{1/4} \). Once you've turned radicals into exponentials, you can easily perform operations like division using the law of exponents.
To simplify radicals:

• Express the number under the root as a power of its base.
• Convert the root into a fraction exponent.
• Proceed with simplification using standard algebraic rules.

Basics of Law of Exponents

The law of exponents is a set of rules that helps in simplifying expressions involving powers. One of the crucial rules is \( \frac{a^m}{a^n} = a^{m-n} \). This rule is extremely useful when dealing with division of similar bases raised to different powers.
In our expression \( \frac{15 \times 5^{3/4}}{5^{1/4}} \), applying the law of exponents means subtracting the exponent in the denominator from the exponent in the numerator: \( 5^{3/4 - 1/4} = 5^{2/4} = 5^{1/2} \).
Understanding and applying this rule lets you simplify and solve expressions more easily, avoiding lengthy calculations. Remember:

• Keep the base the same, only subtract the exponents.
• Resulting expression shows the power of the base "in the air."
• Helps make complex expression easier to handle.

Simplifying Fractions

Fraction simplification involves making a fraction as simple as possible. This means performing any possible operations to condense the expression into its simplest form. With radicals and exponents, it's all about using previously discussed rules to break down and rebuild the expression into something neat and tidy.
By converting \( 5^{1/2} \) from its fractional format into \( \sqrt{5} \), the resulting expression \( 15 \times \sqrt{5} \) is much simpler and more comprehensible.
To simplify fractions effectively:

• Simplify the numerator and the denominator separately, if possible.
• Use exponent rules to condense the expression.
• Convert back to radicals for a clean, understandable result, especially if the task is instructive or educational.