Problem 15 - Elementary and Intermediate Algebra

Question

Fill in the blanks. $$ \sqrt{x^{2}}=\quad \text { and } \sqrt[3]{x^{3}}= $$

Step-by-Step Solution

Answer

\( \sqrt{x^2} = |x| \) and \( \sqrt[3]{x^3} = x \).

Step 1: Understand the Square Root

The expression \( \sqrt{x^2} \) involves finding the square root of \( x^2 \). For any non-negative number, the square root of its square is the number itself, thus \( \sqrt{x^2} = x \). However, for absolute correctness, taking into account negative values, it is safer to say \( \sqrt{x^2} = |x| \), because square roots return non-negative results.

Step 2: Understand the Cube Root

For \( \sqrt[3]{x^3} \), we are finding the cube root of \( x^3 \). The cube root of a cubed number is simply the number itself, regardless of whether it's positive or negative, so \( \sqrt[3]{x^3} = x \). This is because cubing and then taking a cube root reverse the operation entirely.

Step 3: Final Answer

Fill in the blanks with the expressions obtained from the steps above. The final answers are as follows: \( \sqrt{x^2} = |x| \) and \( \sqrt[3]{x^3} = x \).

Key Concepts

Absolute Value

Understanding absolute value is crucial when working with square roots, especially for expressions like \( \sqrt{x^2} \). The absolute value of a number, denoted \( |x| \), represents its distance from zero on the number line, irrespective of its direction. This distance is always non-negative, hence:

• \( |3| = 3 \) because three units from zero is 3.
• \( |-3| = 3 \) because the distance is still 3, even though it’s in the opposite direction.

When taking the square root of a squared number, such as \( x^2 \), it's essential to express it as \( |x| \) to cover all values of \( x \), irrespective of their sign.
For instance, both \( (2)^2 \) and \( (-2)^2 \) equal 4, hence \( \sqrt{4} = 2 \), which corresponds to the absolute value \( |x| = 2 \). So, when writing \( \sqrt{x^2} \), the correct expression is \( |x| \), ensuring that we account for both positive and negative numbers.

Cube Root

The cube root is represented as \( \sqrt[3]{x} \), and it reverses the cubing of a number. Just as cubing a number involves multiplying it by itself twice (\( x \times x \times x \)), the cube root "un-does" this operation. Unlike square roots, cube roots can result in negative numbers because a negative number raised to an odd power remains negative.
Consider:

• \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \).
• \( \sqrt[3]{-8} = -2 \) because \( (-2)^3 = -8 \).

Thus, for an expression like \( \sqrt[3]{x^3} \), the cube root and cubing action cancel each other, leaving us with \( x \). This outcome is independent of whether \( x \) is positive, negative, or zero, making cube roots an essential tool in dealing with polynomials and algebraic expressions.

Exponents and Radicals

Exponents refer to repeated multiplication of a number by itself, while radicals involve the opposite operation – finding roots. These concepts are foundational for understanding how to simplify and manipulate mathematical expressions.**Understanding Exponents**

• An exponent, like 2 in \( x^2 \), implies \( x \times x \).
• A special term \( x^3 \) means \( x \times x \times x \).

Exponents dictate how many times a number is multiplied, simplifying expressions and allowing for quicker calculations.**Understanding Radicals**

• A radical sign \( \sqrt{} \) signifies a square root, aiming to find the original number when squared gives the present result.
• A cube root, represented by \( \sqrt[3]{} \), does the same for cubing.

Radicals unwind exponents. Together, they create a powerful duo in mathematics. For example, when you see \( \sqrt{x^2} \), you know you're taking the square root of the squared value, similarly \( \sqrt[3]{x^3} \) gives the original number itself as they cancel each other.