Problem 137 - Elementary and Intermediate Algebra

Question

Explain why \(\sqrt{m} \cdot \sqrt{m}=m\) but \(\sqrt[3]{m} \cdot \sqrt[3]{m} \neq m .\) Assume that \(m\) represents a positive number.

Step-by-Step Solution

Answer

The square results in the original number \(m\), but two cube roots only produce \(\sqrt[3]{m^2}\).

Step 1: Understanding the Basics

We start by understanding the initial expressions: \ \(\sqrt{m} \cdot \sqrt{m}\) and \(\sqrt[3]{m} \cdot \sqrt[3]{m}\), where the first involves square roots and the second involves cube roots. Assuming \(m\) is a positive number, we will analyze how these results differ based on the properties of roots.

Step 1: Evaluating Square Roots

For the expression \(\sqrt{m} \cdot \sqrt{m}\), we can apply the property of square roots: \ \(\sqrt{a} \cdot \sqrt{a} = a\). This means that \(\sqrt{m} \cdot \sqrt{m} = m\). This follows directly from the definition of the square root, which states that the square root of a number \(m\) is a value that, when multiplied by itself, gives \(m\).

Step 2: Evaluating Cube Roots

Now consider \(\sqrt[3]{m} \cdot \sqrt[3]{m}\). The cube root property indicates that \(\sqrt[3]{a} \cdot \sqrt[3]{a} eq a\); instead, it simplifies to \(\sqrt[3]{a^2}\). Therefore, \(\sqrt[3]{m} \cdot \sqrt[3]{m} = \sqrt[3]{m^2}\) and not \(m\).

Step 3: Analyzing the Results

The reason for the difference is that the square root is based on the power of 2: \ \((\sqrt{m})^2 = m\), while the cube root involves a power of 3. Thus, multiplying two cube roots doesn't reach \(m\) unless raised a third time: \ \((\sqrt[3]{m})^3 = m\). Therefore, \(\sqrt[3]{m} \cdot \sqrt[3]{m}\) doesn't equal \(m\) but equals \(\sqrt[3]{m^2}\).

Key Concepts

Understanding Square Roots

Square roots are quite intuitive once we break them down. When we say "square root" of a number, it essentially means we are looking for a number that, when multiplied by itself, gives us the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. This is the essence of a square root.

When applied to algebraic expressions, the property becomes:

• If you multiply the square root of any number by itself, you get the original number: \( \sqrt{m} \cdot \sqrt{m} = m \).

This concept is built on exponents, specifically the power of 2. The square root of a number is actually the same as raising it to the power of 1/2. So, when you multiply two square roots together, you're essentially performing this operation: \( (m^{1/2}) \cdot (m^{1/2}) = m^{1/2 + 1/2} = m^1 = m \).

This straightforward relationship makes square roots relatively easy to work with, especially in algebraic expressions.

Exploring Cube Roots

Cube roots take us to a slightly more complex domain compared to square roots. A cube root of a number is a special value, which when used three times in a multiplication, gives that number anew. For example, the cube root of 8 is 2 because 2 times 2 times 2 equals 8. Cube roots are linked to the power of 3, rather than 2.

In algebraic terms:

• The cube root of a number \( m \) is represented as \( \sqrt[3]{m} \).
• If you multiply a cube root by itself, you don’t return directly to \( m \). Instead, the outcome is \( \sqrt[3]{m^2} \).

This relationship illustrates why \( \sqrt[3]{m} \cdot \sqrt[3]{m} eq m \); you are merely at the squared stage in terms of exponents because \( (m^{1/3}) \cdot (m^{1/3}) = m^{1/3 + 1/3} = m^{2/3} \).

To equate back to \( m \), you would need one more multiplication by \( \sqrt[3]{m} \), completing the cube.

Properties of Roots in Algebraic Expressions

Roots are a fundamental concept in algebra, allowing us to inverse exponentiation and work with different powers. Understanding their properties allows you to manipulate and simplify algebraic expressions more effectively.

Here are some essential properties to consider:

• Product Property of Roots: This states that the root of a product is the product of the roots, \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).
• Quotient Property of Roots: Similarly, the root of a quotient is the quotient of the roots, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), provided \( b eq 0 \).
• Power under Root: This describes how roots work with powers, such as \( (\sqrt[n]{a^m} = a^{\frac{m}{n}}) \). This is critical in decomposing higher-order roots into manageable forms.

These properties are invaluable when working with algebraic expressions involving square and cube roots, promoting simplification and understanding of these expressions. Mastering them paves the way to tackle more complex problems effectively.