Problem 32 - Algebra A Combined Function

Question

Find each cube root. $$ \sqrt[3]{x^{15}} $$

Step-by-Step Solution

Answer

\(\sqrt[3]{x^{15}} = x^5\).

Step 1: Understand the problem

We need to find the cube root of \(x^{15}\). This can be written as \((x^{15})^{1/3}\). The goal is to simplify this expression.

Step 2: Apply the Power of a Power Rule

When raising a power to another power, we multiply the exponents according to the rule \((a^m)^n = a^{m\cdot n}\). Here, \((x^{15})^{1/3}\) becomes \(x^{15 \cdot (1/3)}\).

Step 3: Multiply the Exponents

Multiply the exponents: \(15 \cdot \frac{1}{3} = 5\). Therefore, \((x^{15})^{1/3} = x^{5}\).

Step 4: Verify the Result

Check that \(x^5\) is the correct simplification by raising it back to the power of 3: \((x^5)^3 = x^{5\cdot 3} = x^{15}\), which matches the original expression.

Key Concepts

Exponent Rules

Exponents are a powerful way to represent repeated multiplication. Consider the expression \(x^n\), which means \(x\) is multiplied by itself \(n\) times. For example, \(x^3\) implies \(x \cdot x \cdot x\). This concept is foundational to algebra and helps to simplify complex expressions.
When dealing with exponents, there are some basic rules that can help simplify the expressions:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\). Add the exponents if the bases are the same.
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\). Subtract the exponents when dividing.
• Zero Exponent Rule: \(a^0 = 1\) for any \(a eq 0\).

Understanding these rules makes it easy to manipulate and simplify expressions that include exponents. By knowing which rule to apply, you can reduce a seemingly complicated expression to a more manageable form. This is crucial when working with powers and roots simultaneously. Remember, mastering these rules will greatly enhance your algebraic skills.

Power of a Power Rule

The power of a power rule is a specific application within exponent rules that helps simplify expressions where exponents are involved. This rule states that when you have an expression like \((a^m)^n\), you can simplify it by multiplying the exponents: \(a^{m \cdot n}\).
This rule is particularly useful when working with radical expressions, such as cube roots. For example, finding the cube root of \(x^{15}\) can be expressed as \((x^{15})^{1/3}\). Applying the power of a power rule, this becomes \(x^{15 \cdot \frac{1}{3}} = x^5\).
Being able to quickly and correctly apply this rule allows for easier manipulation of expressions. It reduces a complex chain of multiplication to a much simpler form. Understanding the power of a power rule ensures you aren't intimidated by expressions that involve several layers of exponents. This efficiency is beneficial in solving not only algebra problems but also complex calculus equations where simplification is key.

Simplifying Radicals

Radicals involve the root of a number or expression, usually indicated by the radical sign \(\sqrt{}\), but when dealing with cube roots, we often see \(\sqrt[3]{}\). Simplifying radicals means finding an expression that is equivalent but in a simpler form.
For cube roots, you are looking for a number or expression that, when multiplied by itself three times, returns the original value. For example, to simplify \(\sqrt[3]{x^{15}}\), you first convert it into an expression with exponents: \((x^{15})^{1/3}\).
By applying exponent rules, specifically the power of a power rule as discussed, you can reshape it into \(x^5\). This version of the expression has the same value in a much cleaner and shorter form. Simplifying radicals is a useful skill as it declutters expressions, making them easier to work with in equations or when graphing functions. Always aim for the simplest version to keep calculations straightforward.