Problem 122 - Intermediate Algebra

Question

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[5]{x^{49}} $$

Step-by-Step Solution

Answer

\( x^9 \times \sqrt[5]{x^4} \)

Step 1: Apply the Power Rule for Radicals

The expression \( \sqrt[5]{x^{49}} \) can be simplified using the power rule for radicals, which states that \( \sqrt[n]{a^m} = a^{m/n} \). Applying this rule gives us \( x^{49/5} \).

Step 2: Simplify the Exponent

Now, simplify the exponent \( 49/5 \). Divide 49 by 5, which gives us the quotient 9 and a remainder of 4, or equivalently \( 49/5 = 9 + 4/5 \). This can be represented as \( x^9 \times x^{4/5} \).

Step 3: Rewrite in Radical Form

The term \( x^{4/5} \) can be rewritten in radical form as \( \sqrt[5]{x^4} \). Thus, the expression becomes \( x^9 \times \sqrt[5]{x^4} \).

Key Concepts

Power Rule for Radicals

When simplifying expressions involving radicals, the power rule for radicals is a very useful tool. This rule essentially allows you to convert a radical expression into one involving exponents. The power rule for radicals states that \( \sqrt[n]{a^m} = a^{m/n} \). This means you can transform a radical expression into an exponent where the index of the radical becomes the denominator of the exponent, and the power of the radicand becomes the numerator.
This conversion streamlines the process of simplification, especially when dealing with large powers. For example, given an expression like \( \sqrt[5]{x^{49}} \), applying the power rule means rewriting this as \( x^{49/5} \).
This transformation often leads to more straightforward calculations and further simplification possibilities.

Exponent Rules

Exponent rules are foundational principles that guide us in handling expressions involving powers. These rules help us manipulate exponents easily. With the expression \( x^{49/5} \), understanding the exponent rules is crucial for breaking down the compound fraction.

• Product of Powers Rule: When multiplying terms with the same base, add the exponents. For example, \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing terms with the same base, subtract their exponents, i.e., \( a^m / a^n = a^{m-n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).

In the original example, the exponent of \( x \) is \( 49/5 \). Breaking this down into \( 9 + 4/5 \) allows us to use the product of powers rule to express it as \( x^9 \times x^{4/5} \). Thus, exponent rules help simplify complex expressions in a systematic and efficient manner.

Fractional Exponents

Fractional exponents are another way to express roots, making it easier to perform calculations and simplify expressions. A fractional exponent like \( a^{m/n} \) represents the \( n \)-th root of \( a^m \). For instance, \( x^{4/5} \) translates to \( \sqrt[5]{x^4} \). This understanding bridges the gap between exponents and radicals, providing a versatile tool in algebra.
When handling fractional exponents, employing exponent rules also becomes important. We express \( x^{49/5} \) as \( x^9 \times x^{4/5} \), allowing us to separate the integer part of the exponent from the fractional part. The integer component \( x^9 \) remains as a simple power, while the fractional component \( x^{4/5} \) converts back into a radical \( \sqrt[5]{x^4} \).
By recognizing the relationship between radicals and fractional exponents, we create a clearer path towards simplifying and manipulating complex algebraic expressions.