Problem 5 - Algebra 2

Question

Simplify. $$ \sqrt[3]{x^{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Cube Root

The expression \( \sqrt[3]{x^3} \) represents the cube root of \( x^3 \). The cube root of a number is a value that when multiplied by itself three times gives the original number.

Step 2: Apply the Cube Root Property

According to the properties of exponents, \( \sqrt[3]{x^3} = x^{3/3} \). This is because the cube root is the same as raising the expression to the power of \( 1/3 \).

Step 3: Simplify the Exponent

Simplity the expression \( x^{3/3} \). Divide the exponent 3 by 3, which simplifies to \( x^1 \).

Step 4: Write the Final Answer

Since any number raised to the power of 1 is itself, \( x^1 = x \). Thus, \( \sqrt[3]{x^3} = x \).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. When you see a number or variable with an exponent, it means you multiply the base by itself as many times as the exponent indicates. For example, in the term \( x^3 \), \( x \) is the base and 3 is the exponent, meaning you multiply \( x \) by itself three times: \( x \times x \times x \).
Understanding exponents is crucial because they allow you to simplify expressions and solve equations more efficiently. They are frequently used in algebra and higher-level mathematics.
When dealing with exponents, there are a few important rules to remember:

• Multiplying Powers: \( x^a \times x^b = x^{a+b} \)
• Dividing Powers: \( x^a / x^b = x^{a-b} \)
• Power of a Power: \((x^a)^b = x^{a \times b} \)
• Zero Exponent: \( x^0 = 1 \) (for \( x eq 0 \))

These rules are part of the foundational knowledge that helps with simplifying and manipulating expressions in algebra and calculus.

Simplification

Simplification in mathematics means reducing an expression to its most basic form. It makes equations easier to handle and solve.
In our exercise, simplifying \( \sqrt[3]{x^3} \) involves utilizing our understanding of exponents and roots. By recognizing that the cube root is an inverse operation to cubing, the simplification process reduces the expression to a simpler form.
The trick is to use the rule that says the nth root of a number raised to the nth power returns the original number. This means the cube root of \( x^3 \) is \( x \). To see why this works, consider:

• \( \sqrt[n]{a^n} = a \), where n is a positive integer.
• For this example, \( n = 3 \), so \( \sqrt[3]{x^3} = x \).

We apply properties of exponents, showing that \( x^3 \times (1/3) = x^{3/3} = x^1 = x \). Therefore, simplifying means finding this base value without losing its mathematical properties.

Properties of Exponents

Properties of exponents are the rules that govern how exponents can be manipulated. Understanding these properties allows for consistent simplification and solving of algebraic expressions.
In the context of our exercise, these properties are pivotal for reducing expressions effectively. For instance, the property used in finding the cube root is directly related to the division of exponents:

• When taking the cube root of \( x^3 \), we translate it to an exponent form as \( x^{3 \times (1/3)} \).
• This operation makes use of the Power of a Fraction: \( a^{b/c} = \sqrt[c]{a^b} \).
• Simplifying \( x^{3/3} \) results from the Division of Exponents property: \( x^a / x^b = x^{a-b} \).

By dividing the exponents (in this case, 3 by 3), we are left with the simpler form, \( x^1 \), which simplifies further to \( x \).
These exponent properties are crucial, not only in simplification but also in diverse mathematical computations such as solving equations and modeling real-world scenarios.